Divided differences

Română

It is known that the divided difference of order \(n\) of a function \(f\) on the simple nodes \(x_0, x_1, \ldots, x_n\), which is denoted by

is defined by a recurrence formula, starting from the definition of the divided difference of order one on the nodes \(x_0, x_1\).

In the present work, the divided difference of order \(n\) on the simple nodes \(x_0, x_1, \ldots, x_n\), the notation being chosen so that \(x_0 \lt x_1 \lt \ldots \lt x_n\), is defined in the first place by means of a boundary value problem. To this end we use “the method of the function \(\varphi\)”, which assumes that \(f \in C^n[x_0, x_n]\), and we arrive at the formula

The function \(\varphi\) has been studied, and it has been shown that it is positive on the interval \((x_0, x_n)\), which is a fundamental result of the work.

The definition of the divided difference of order \(n\) on simple nodes by means of a boundary value problem is extended, again by means of a boundary value problem, to the case of the nodes \(x_0, x_1, \ldots, x_k\), where \(x_0 \lt x_1 \lt \ldots \lt x_k\), of multiplicities \(n_0, n_1, \ldots, n_k\), and this divided difference is denoted by

Denoting by \(n\) the number defined by \(n_0 + n_1 + \ldots + n_k = n + 1\) and assuming that \(f \in C^n[x_0, x_k]\), we prove the formula

which gives at once the definition of the divided difference on multiple nodes and its integral representation. As in the case of formula (1), it is shown that the function \(\varphi\) is positive on the interval \((x_0, x_k)\), the proof, however, being much more delicate than in the case of simple nodes.

Applications are made of divided differences on simple or multiple nodes, and of their integral representations, to the Lagrange and Lagrange–Hermite interpolation formulas and to their remainders. We also determine the remainders of the Gauss and Turán quadrature formulas, establishing their connection with the integral representation of certain divided differences on multiple nodes.

The interest of the work also lies in the extension of the method of the function \(\varphi\) to the definition of the divided difference of order \(n\) of a function of two variables on the grid of nodes \(M_{i,k}\), \((i, k = 0, 1, \ldots n)\), which is denoted by

and where \(x_0 \lt x_1 \lt \ldots \lt x_n\); \(y_0 \lt y_1 \lt \ldots \lt y_n\).

Under certain hypotheses on the function \(f\), which have been stated precisely in the work, it has been shown that we have the integral representation

where \(D\) is the rectangle \(x_0 \leq x \leq x_n\); \(y_0 \leq y \leq y_n\), and the function \(\varphi\) has been studied, it being shown that it is positive in the open rectangle \(D\) and vanishes on its boundary.

We have likewise extended the definition of the divided difference of order \(n\) to a function \(f\) of three variables on the grid of nodes \(M_{i,j,k}(i, j, k = 0, 1, \ldots, n)\), where \(x_0 \lt x_1 \lt \ldots \lt x_n\); \(y_0 \lt y_1 \lt \ldots \lt y_n\); \(z_0 \lt z_1 \lt \ldots \lt z_n\), which is denoted by

Under certain hypotheses on the function \(f\), which have been stated precisely in the work, it has been shown that we have the integral representation

where \(D\) is the parallelepiped \(x_0 \leq x \leq x_n\); \(y_0 \leq y \leq y_n\); \(z_0 \leq z \leq z_n\).

The function \(\varphi\) is positive in the open parallelepiped \(D\) and vanishes on its boundary.

Between the divided differences of order \(n\) of a function of \(p\) variables there is a recurrence relation, which has been established for \(p = 2, 3\). Taking it as the definition in general, and making suitable hypotheses on the function \(f\), we have established the definition of the divided difference of order \(n\) of a function \(f\) of the variables \(x_1, x_2, \ldots, x_p\) on the grid of nodes \(M_{i_1, i_2, \ldots, i_p}\), which we have denoted by

where \(x_h^{(0)} \lt x_h^{(1)} \lt \ldots \lt x_h^{(n)}\), \(h = 1, 2, \ldots, p\).

We have shown that the following integral representation holds:

where \(D\) is the domain \(x_1^{(0)} \leq x_1 \leq x_1^{(n)}\); \(\ldots\); \(x_p^{(0)} \leq x_p \leq x_p^{(n)}\).

It has been proved that the function \(\varphi(x_1, x_2, \ldots, x_p)\) is positive in the open domain \(D\) and vanishes on its boundary.

D. V. Ionescu

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Română

For 25 years I have been publishing papers on numerical analysis regularly. I use a very fruitful research method, which I have called the method of the function \(\varphi\).

This method is essentially Green’s method, well known in mathematical physics, applied here to problems of numerical analysis. Several authors have published papers with concerns of this kind, among whom I mention H. Radon (1935) [33], G. Kowalewski (1932) [23], A. Ghizzetti and A. Ossicini (1970) [27].

The aim of the present work is, in the first place, to set out the definition of the divided difference of order \(n\) of a function \(f\) of one variable on the nodes \(x_0, x_1, \ldots, x_n\), where \(x_0 \lt x_1 \lt \ldots \lt x_n\), which is denoted by \([x_0, x_1, \ldots x_n; f]\), and to give its integral representation when \(f \in C^n[x_0, x_n]\).

This is done by means of the method of the function \(\varphi\), by treating a suitably posed boundary value problem which leads to the definition of the divided difference and to its integral representation by the formula

We shall further specify the function \(\varphi\) and show that the research method employed allows it to be studied on the interval \([x_0, x_n]\).

We shall then treat the consequences that follow from this method for obtaining formula (1), and the most important part of the work will be devoted to the definition of the divided difference of order \(n\) for functions of two variables on a grid \(M_{i,k}(x_i, y_k)\), \(i, k = 0, 1, \ldots, n\), and to its integral representation, as well as for functions of three variables on a grid \(M_{i,j,k}(x_i, y_j, z_k)\), \(i, j, k = 0, 1, \ldots, n\), together with its integral representation.

The working method employed in the case of \(n\) variables, \(n = 2\) and \(n = 3\), gives rise to computational difficulties which have been skilfully circumvented and which become more complicated for \(n \gt 3\).

By means of the hypotheses imposed on the function \(f\) in the cases \(n = 2\) and \(n = 3\), we have brought to light a recurrence formula for divided differences.

The recurrence formula was given in the case \(n = 2\), a priori, by T. Popoviciu [31], who did not go further and did not give the integral representation of the divided difference. By changing the hypotheses on the function \(f\), we have arrived at the definition of the divided difference for functions of \(p\) variables by means of a recurrence formula, and at their integral representation. This is the subject of the last chapter of the work.

We shall now give, one by one, an account of the problems treated in each chapter.

1. In order to arrive at formula (1) by applying the method of the function \(\varphi\), we attach to the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{n-1}, x_n]\), respectively, the polynomials \(\varphi_1, \varphi_2, \ldots \varphi_n\), solutions of the differential equations

and we assume that \(f \in C^n[x_0, x_n]\). To each integral \(\int_{x_{i-1}}^{x_i} \varphi_i f^{(n)}\, ds\), \((i = 1, 2, \ldots, n)\), we apply the generalized integration by parts formula, and we add all these formulas. We obtain a formula of the form

where the function \(\varphi\) coincides with the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) on the respective intervals, and where \(\mathscr{P}[f]\) is a functional depending on the values of the function \(f\) and of its derivatives \(f', f'', \ldots, f^{(n-1)}\) at the nodes. In connection with formula (3) we pose the following:

The solution \(\varphi\) of this problem is the spline function of order \(n-1\) defined on the interval \([x_0, x_n]\) with the nodes \(x_0, x_1, \ldots, x_n\) [35].

Replacing in formula (3) the function \(\varphi\) by the solution of the boundary value problem, we obtain the integral representation of the divided difference \([x_0, x_1, \ldots, x_n; f]\), that is, formula (1).

The function \(\varphi\), the solution of the boundary value problem posed above, is of great importance in the study of the divided difference. By means of the boundary conditions of the problem it is proved that the function \(\varphi\) is positive on the interval \((x_0, x_n)\). Formula (1) has also been obtained by other mathematicians, among whom I mention G. Peano [30], A. Sard [34], L. Tchakaloff [46].

In subsection 1.1.3 I have given a proof of formula (1) which arose from a conversation with Prof. P. J. Laurent [24] and which I attribute to Prof. Laurent.

I have applied formula (1) to establishing the Lagrange interpolation formula and its remainder, as well as to the theory of differences.

2. In Chapter II I have applied the method of the function \(\varphi\) in order to define the divided difference of a function \(f\) on the nodes \(x_0, x_1, \ldots, x_k\), \(x_0 \lt x_1 \lt \ldots \lt x_k\), of multiplicities \(n_0, n_1, \ldots, n_k\), and to obtain its integral representation.

Defining the number \(n\) by the formula \(n + 1 = n_0 + n_1 + \ldots + n_k\), I have attached to the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{k-1}, x_k]\) the polynomials \(\varphi_1\; \varphi_2, \ldots, \varphi_k\), solutions of the differential equations

Assuming that \(f \in C^n[x_0, x_k]\), I have applied the generalized integration by parts formula to the integrals \(\int_{x_{i-1}}^{x_i} \varphi_i f^{(n)}\, ds\), for \(i = 1, 2, \ldots, k\), and have added all these formulas term by term. I have obtained a formula of the form

where \(\mathscr{P}[f]\) is a functional depending on the values of \(f, f', f'', \ldots, f^{(n-1)}\) at the nodes \(x_0, x_1, \ldots, x_k\), while the function \(\varphi\) coincides with the polynomials \(\varphi_1, \varphi_2, \ldots \varphi_k\) on the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{k-1}, x_k]\).

In connection with formula (5) I have treated the following:

It is proved that this boundary value problem has a unique solution.

If we denote by \(\Delta[f]\) the determinant (2.1.22)

then

Replacing in formula (5) the function \(\varphi\) by the solution of the boundary value problem, we obtain the integral representation of the divided difference (6), namely

By means of the boundary conditions of the boundary value problem, and with the preparation made in § 2.2 and § 2.3, there is proved in § 2.4 the fundamental theorem that the function \(\varphi\) in the integral representation (9) is positive on the interval \((x_0, x_k)\).

The proof I gave to show that the function \(\varphi\) in the integral representation (9) is positive on the interval \((x_0, x_k)\) when the nodes are multiple was used by Paraschiva Pavel to prove that the function \(\varphi\) in the integral representation of the remainder of the Gauss-type quadrature formula with multiple but distinct nodes, given by Tiberiu Popoviciu, keeps a constant sign on the interval of integration.

In § 2.5 the divided difference of a function on multiple nodes and its integral representation are given an important application: obtaining the Lagrange–Hermite interpolation formula and studying its remainder.

3. In Chapter III, § 3.2, a connection is established between the Gauss quadrature formula [9]

where the nodes \(x_1, x_2, \ldots, x_n\) are the roots of the Legendre polynomial associated with the interval \([a, b]\) and \(f \in C^{2n}[a, b]\), and the divided difference of a function \(g \in C^{2n+1}[a, b]\):

Writing

we have the following theorem:

Since \(A_0 + A_{n+1} = 0\) and \(A_{n+1} \neq 0\), we deduce:

4. In § 3.3 we study P. Turán’s quadrature formula [47]

where \(p\) is a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\), while the right-hand side contains the derivatives \(f^{(l)}\) for \(l = 0, 1, \ldots, 2(s-1)\) at the nodes \(x_1, x_2, \ldots, x_n\).

This problem is connected with a minimum problem of D. Jackson [22], namely: to determine uniquely the coefficients of the polynomial

such that the integral

where \(s\) is a natural number, is minimal.

In this work we give the proof of this theorem due to Margareta Frenkel [4], which seems to us better suited to the problems of Numerical Analysis than D. Jackson’s proof.

The polynomial \(\omega_n(x)\) which realizes the minimum of the integral (15) is called the minimizing polynomial.

It is proved that the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the minimizing polynomial are uniquely determined by the equations

The minimizing polynomial \(\omega_n\) has all its roots real, distinct and lying between \(a\) and \(b\).

Once the nodes in formula (13) have been determined, the coefficients \(C_1^{(l)}, \ldots, C_n^{(l)}\) for \(l = 0, 1, \ldots, 2(s-1)\) are easily determined.

5. Let us assume that \(p(x) = 1\). The remainder of Turán’s quadrature formula is then determined by a method analogous to the one presented in item 3 for determining the remainder of the Gauss formula.

Let us consider the divided difference

of a function \(g \in C^{2ns+1}[a, b]\). We can write:

Since \(A_0 + A_{n+1} = 0\) and \(A_{n+1} \neq 0\), we deduce:

6. In § 3.4 and § 3.5 the Gauss and P. Turán quadrature formulas with the weight \(\pi(x)\) are extended to the case when the multiple nodes \(a\) and \(b\), of multiplicities \(p\) and \(q\), are added to the nodes \(x_1, x_2, \ldots, x_n\).

When the weight is \(\pi(x) = 1\), one proceeds as in item 4, showing that the remainders of the Gauss and Turán quadrature formulas are connected with the integral representation of the divided differences (11) and (19), in which, however, the nodes \(a\) and \(b\) are multiple, of multiplicities \(p + 1\) and \(q + 1\) [11].

7. In Chapter IV the method of the function \(\varphi\), used to define the divided difference of order \(n\) of a function of one variable and to obtain its integral representation, is extended to functions of two variables. This extension will again be reached by means of a boundary value problem. At the basis of this work lies the integration by parts formula for a double integral. In § 4.1 it is shown that this formula was established by M. Nicolescu [26], and we shall write it in a form which we consider convenient for our work.

9. Let us consider the rectangle

the nodes \(M_{i,k}(x_i, y_k)\), where

and the rectangles

Let us further consider the functions \(\varphi_{i,k}\), \((i, k = 0, 1, \ldots, n-1)\), continuous on \(D_{i,k}\) together with the partial derivatives

To each rectangle \(D_{i,k}\) and to the pair of functions \(f, \varphi_{i,k}\) let us apply the fundamental formula (23). We add all these formulas term by term and thus obtain the important formula:

where \(\mathscr{P}[f]\) is written more briefly in the form

where on the right-hand side we write all the terms with the following convention:

10. In connection with formula (26) and with the expression (27) of \(\mathscr{P}[f]\), we pose the following:

For this purpose the following boundary conditions are introduced:

for \(r = 0, 1, \ldots, n-1\) and \(i, k = 0, 1, \ldots, n\). When we take \(i = 0\), \(k = 0\) or \(i = n\), \(k = n\); or \(i = n\); or \(k = n\), convention (A) must be taken into account in the boundary conditions (29). This problem is denoted by \((P_{i,k})\). First the problem \((P_{0,0})\) is solved, then successively \((P_{1,0}), \ldots, (P_{n-1,0})\); \((P_{0,1}), \ldots, (P_{n-1,1}), \ldots\); and so on.

The constants \(C_{i,k}\) will be determined up to a factor \(H\), so that the boundary value problem is possible. The factor \(H\) is determined so that

With skill and patience one finds that

where the constants \(C_{i,k}\) are determined by also taking into account the last condition (30), and we have

If in \(\mathscr{P}[f]\) we replace the functions \(\varphi_{i,k}\) by the solution of the boundary value problem, we find

where \(\varphi\) is the solution of the boundary value problem treated in § 4.2.

11. Taking into account the expression of the coefficients \(C_{i,k}\), introducing the divided differences of order \(n\) of the function \(u\) on the nodes \(x_0, x_1, \ldots, x_n\) and of the function \(v\) on the nodes \(y_0, y_1, \ldots, y_n\), and assuming that \(u \in C^n[x_0, x_n]\), \(v \in C^n[y_0, y_n]\), we have

The following theorems result:

It follows that the function \(\varphi\) in the integral representation (36) is the product of the spline functions \(\psi\) and \(\theta\) in the integral representations (37), (38).

12. Chapter V deals with the definition of the divided difference of a function \(f\) of three variables on a grid of nodes \(M_{i,j,k}(x_i, y_j, z_k)\), where \(i, j, k = 0, 1, \ldots, n\), and gives its integral representation under certain hypotheses on the function \(f\).

13. The generalized integration by parts formula for triple integrals.

14. Let us consider the parallelepiped \(D\) defined by the inequalities

and the grid of nodes \(M_{i,j,k}\), \((i, j, k = 0, 1, \ldots, n)\), with coordinates \((x_i, y_j, z_k)\), where

We shall denote by \(D_{i,j,k}\) the parallelepiped defined by the inequalities \(D_{i,j,k}\):

where \(i, j, k = 0, 1, \ldots, n - 1\).

With each parallelepiped \(D_{i,j,k}\) we associate a function \(\varphi_{i,j,k}\) continuous on \(D_{i,j,k}\) together with its partial derivatives appearing in the sequence (44). We likewise associate with the parallelepiped \(D\) a function \(f\) continuous on \(D\) together with the partial derivatives appearing in the sequence (44). Then the generalized integration by parts formula for triple integrals (45) can be applied to each parallelepiped \(D_{i,j,k}\) and to the functions \(f\), \(\varphi_{i,j,k}\). Adding all these formulas term by term, we shall have the formula

where we have denoted by \(\mathscr{P}[f]\) the functional resulting from the computation, that is

In applying formula (48) one takes into account:

15. In connection with formulas (47) and (48) we shall treat the following:

For this purpose we shall introduce the following boundary conditions:

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Next

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Finally

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

The constants \(C_{i,j,k}\) will be determined up to a factor, which is determined by a new condition, namely

When one of the indices in \(\varphi_{i,j,k}\) is replaced by \(-1\) or \(n\), convention (A) must be taken into account.

16. The solution of this problem is

It is proved that the constants \(C_{i,j,k}\) are given by the following formulas:

The expression of \(\mathscr{P}[f]\), when the solution of the boundary value problem is put in place of the functions \(\varphi_{i,j,k}\), is

where

We have several theorems:

We have

18. If for the functions \(u \in C^n[x_0, x_n]\), \(v \in C^n[y_0, y_n]\), \(w \in C^n[z_0, z_n]\) and their divided differences on the nodes \(x_0, x_1, \ldots, x_n\), \(y_0, y_1, \ldots, y_n\), \(z_0, z_1, \ldots, z_n\) we have the integral representations

formula (61) shows that

20. T. Popoviciu defined the divided difference of order \(n\) of a function of two variables on the grid of nodes \(M_{i,k}(x_i, y_k)\), \((i, k = 0, 1, \ldots, n)\), by the recurrence formula, which he gave a priori,

T. Popoviciu was not concerned with the integral representation of the divided difference (65).

Formula (65) also follows from our formulas (32), (33), obtained with the boundary value problem treated in Chapter IV and the hypotheses of that chapter. In addition, the integral representation of the divided difference (65) has also been obtained, by formula (36).

From the boundary value problem treated in Chapter V there resulted the definition of the divided difference of a function \(f\) of three variables on the grid of nodes \(M_{i,j,k}(x_i, y_j, z_k)\), \((i, j, k = 0, 1, \ldots, n)\), by formula (59), and its integral representation by formula (60), under the hypotheses of Chapter V imposed on the function \(f\).

From formula (59) there follows the recurrence formula

Formula (66) is important; it gives us an idea of how one could define the divided difference of a function \(f\) of \(p\) variables on a grid of nodes \(M_{i_1, i_2, \ldots, i_p}\), and of how its integral representation can be obtained, by making suitable hypotheses on the function \(f\), different from those made in Chapters IV and V, in order to treat the corresponding boundary value problems.

21. Let us consider a function \(f\) of the variables \(x_1, x_2, \ldots, x_p\), and let us take on the axis \(Ox_h\), which corresponds to the variable \(x_h\), the nodes \(x_h^{(0)}, x_h^{(1)}, \ldots, x_h^{(n)}\) such that

Letting \(h\) take the values \(1, 2, \ldots, p\), we obtain a grid of nodes \(M_{i_1, i_2, \ldots, i_p}\) with coordinates \((x_1^{(i_1)}, x_2^{(i_2)}, \ldots, x_p^{(i_p)})\), \(i_1, i_2, \ldots, i_p\) taking values in the set of numbers \(0, 1, \ldots, n\).

Then we have the following definition:

22. The integral representation of the divided difference (67) can be obtained when the function \(f\) satisfies the following hypotheses, which we denote by \((H_p)\), \((H_{p-1})\), \(\ldots\), \((H_1)\).

We consider the domain \(D\) defined by the following inequalities:

Hypothesis \((H_p)\) states that all the partial derivatives

exist and are continuous in \(D\) for \(h = 0, 1, \ldots, n\).

Hypothesis \((H_{p-1})\) states that all the partial derivatives

exist and are continuous in \(D\), for \(h = 0, 1, \ldots, n\).

And so on.

The last hypothesis \((H_1)\) states that all the partial derivatives

exist and are continuous in \(D\) for \(h = 0, 1, \ldots, n\).

This being so, let us consider, in general, the integral representation of the divided difference of order \(n\) of the function \(g\) of the variable \(x_h\) on the nodes \(x_h^{(0)}, x_h^{(1)}, \ldots, x_h^{(n)}\), which, when \(g \in C^n[x_h^{(0)}, x_h^{(n)}]\), is given by the formula

where \(\varphi_h\) is the spline function studied in Chapter I, which is positive on the interval \((x_h^{(0)}, x_h^{(n)})\).

23. We have the following:

24. Several Romanian mathematicians have published books on Numerical Analysis, among whom I mention Acad. Gh. Marinescu [25], Acad. T. Popoviciu [32], D. V. Ionescu [8], C. M. Bucur [1], I. Cuculescu [2], Al. I. Șchiop [37], [38] and Mărgărit Pavelescu [29]; these books have been very well received among mathematicians.

I hope that this book, Divided Differences, which aims to introduce a systematic method of research into Numerical Analysis, the method of the function \(\varphi\), will also contribute to a still greater development of research in this field.

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Română

§ 1.1. The first boundary value problem

In this section we shall study a boundary value problem which will lead to the definition of the divided difference of order \(n\) of a function \(f\) on the simple nodes \(x_0, x_1, \ldots, x_n\), where \(x_0 \lt x_1 \lt \ldots \lt x_n\), and to its integral representation when \(f \in C^n[x_0, x_n]\).

1.1.1. Let us consider the simple nodes \(x_0, x_1, \ldots, x_n\), where

and a function \(f \in C^n[x_0, x_n]\).

To the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{n-1}, x_n]\) let us attach the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) of degree at most \(n - 1\), which means that

and let us compute the integrals

Applying the generalized integration by parts formula to each integral, we shall have

Adding formulas (1.1.4) term by term, we shall obtain a formula whose right-hand side, which we denote by \(\mathscr{P}[f]\), depends on the values of \(f, f', \ldots, f^{(n-1)}\) at the nodes \(x_0, x_1, \ldots, x_n\). We shall thus have the formula

where the function \(\varphi\) coincides with the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) on the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{n-1}, x_n]\).

In connection with formula (1.1.5) we pose the following

The polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) must therefore satisfy the boundary conditions

and, in addition, we must have

for \(f(x) = x^n\).

We shall prove that the boundary value problem thus posed has a unique solution. Then, replacing the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) in formula (1.1.4) by the solution of this problem and adding, we obtain a formula of the form (1.1.5), which is very important and which is the subject of this section.

1.1.2. It is evident that the polynomials

satisfy the differential equations (1.1.2) and the boundary conditions (1.1.6) relative to the nodes \(x_0, x_1, \ldots, x_{n-1}\), whatever \(\lambda_0, \lambda_1, \ldots, \lambda_{n-1}\) may be. The constants \(\lambda_0, \lambda_1, \ldots, \lambda_{n-1}\) will be determined so that the boundary conditions relative to the node \(x_n\) are satisfied as well.

We shall have the system of equations

which determines \(\lambda_0, \lambda_1, \ldots, \lambda_{n-1}\) up to a factor.

Introducing a new constant \(\lambda_n\) by the equation

the system of equations (1.1.9) and equation (1.1.10) reduce to the system

which determines \(\lambda_0, \lambda_1, \ldots, \lambda_n\) up to a factor.

The solution of system (1.1.11) is given by the formulas

where \(V(x_0, x_1, \ldots, x_n)\) is the Vandermonde determinant and where \(H\) is a new constant which will be determined presently.

The polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) are thus determined by formulas (1.1.8) up to a factor \(H\), and therefore we can use them.

We shall first have

Replacing the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) in formulas (1.1.4) by the solution of the boundary value problem and adding, we shall have

that is,

Taking formulas (1.1.12) into account, we can write

that is,

Now let us determine the constant \(H\). Replacing the function \(f(x)\) in formula (1.1.15) by \(x^n\) and taking condition (1.1.7) into account, we shall have \(H \cdot V(x_0, x_1, \ldots, x_n) = 1\), and therefore

The constant \(H\) having been determined, formula (1.1.15) becomes

Formula (1.1.5), in which the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) are replaced by the solution of the boundary value problem posed in subsection 1.1.1, can be written, by means of formula (1.1.17) and notation (1.1.18), in the form

and we have thus arrived at the following fundamental theorem:

Taking formula (1.1.14) into account, formula (1.1.19) can also be written in the form

where the coefficients \(\lambda_i\) are given by formulas (1.1.12) and (1.1.16), that is,

One can also write

and there follows the identity

which gives a practical means of computing the coefficients of formula (1.1.20).

Formulas (1.1.19), (1.1.20) and (1.1.23) show that we have the identity

Introducing the notation of spline functions

we can write the function \(\varphi\) in the form

because, assuming successively that \(s \in [x_0, x_1]\), \(s \in [x_1, x_2]\), \(\ldots\), \(s \in [x_{n-1}, x_n]\), we recover formulas (1.1.8).

Following I. J. Schoenberg — [35] — the function given by formula (1.1.26) is called the spline function of order \(n-1\) defined on the interval \([x_0, x_n]\) and on the nodes \(x_0, x_1, \ldots, x_n\).

Thus the boundary value problem posed in subsection 1.1.1 can serve as an introduction to the theory of spline functions.

It may also be observed that formula (1.1.17) shows that Definition 1.1.1, which we have given for the divided difference of order \(n\) of the function \(f\) on the nodes \(x_0, x_1, \ldots, x_n\), agrees with the classical definition.

1.1.3. Theorem 1.1.1 was first given by G. Peano [30], and we also find it in the works of several mathematicians, for example in the works of L. Tchakaloff [46] and A. Sard [34]. In our method of finding formula (1.1.19), the boundary value problem posed in subsection 1.1.1 has led us in a remarkable way, at the same time, to Definition 1.1.1 of the divided difference of order \(n\) of the function \(f\) on the nodes \(x_0, x_1, \ldots, x_n\), and to its representation by a definite integral, by formula (1.1.19), when \(f \in C^n[x_0, x_n]\).

Professor P. J. Laurent [24] communicated to me, in a conversation, another proof of formula (1.1.19), which starts from the classical definition of the divided difference of a function and uses Taylor's formula and the symbol \((s - x_i)_+^m\) introduced by formula (1.1.25).

Assuming that \(x_0 \lt x_1 \lt \ldots \lt x_n\) and that \(f \in C^n[x_0, x_n]\), we can apply Taylor's formula and obtain

It follows that we can write the divided difference \([x_0, x_1, \ldots, x_n; f]\), according to its classical definition, in the form

that is,

By means of the notation \((s - x_i)_+^{n-1}\), the preceding formula becomes

or again

Expanding the determinant on the right-hand side along the elements of the last column, we obtain formula (1.1.19), where the function \(\varphi(s)\) is given by formula (1.1.26).

1.1.4. The boundary conditions of the problem posed in subsection 1.1.1 make it possible to study the function \(\varphi\) in formula (1.1.19), whose expression is given by formula (1.1.26). For this purpose we shall give several theorems:

This theorem is evident for \(k = 0\) and \(k = n-1\), and therefore for \(n = 2\). Indeed, by formula (1.1.21), for \(k = 0\) we have \(\lambda_0 \gt 0\), and for \(k = n-1\) we have \((-1)^{n-1}(\lambda_0 + \lambda_1 + \ldots + \lambda_{n-1}) = (-1)^n \lambda_n \gt 0\).

It remains to prove the theorem for \(k = 1, 2, \ldots, n-2\) and \(n \gt 2\).

To this end let us consider the interpolation polynomial \(h_k(x)\) defined by the conditions

It is evident that the polynomial \(h_k(x)\) is of degree at most \(n-1\) and is not a constant, because for \(k = 1, 2, \ldots, n-2\) we have, by conditions (1.1.28), \(h_k(x_0) = 1\), whereas \(h_k(x_{n-1}) = 0\).

Applying Rolle's theorem to the polynomial \(h_k(x)\) and to the intervals \([x_0, x_1]\), \([x_1, x_2]\), \(\ldots\), \([x_{k-1}, x_k]\), \([x_{k+1}, x_{k+2}]\), \(\ldots\), \([x_{n-2}, x_{n-1}]\), we deduce that the derivative \(h_k'(x)\) has \(n-2\) distinct roots, \(k\) to the left of \(x_k\) and \(n-2-k\) to the right of \(x_k\). It follows that the polynomial \(h_k(x)\) has degree \(n-1\).

Indeed, let us suppose that the degree of \(h_k(x)\) were less than \(n-1\). Since the derivative of \(h_k(x)\) vanishes at \(n-2\) distinct points, it would be identically zero, and therefore the polynomial \(h_k(x)\) would be a constant, which is impossible. Hence the degree of the polynomial \(h_k(x)\) is \(n-1\).

It follows that the polynomial \(h_k(x)\) does not vanish for \(x \gt x_{n-1}\), and since it is decreasing on the interval \([x_k, x_{k+1}]\), it has the sign of \((-1)\) on the interval \((x_{k+1}, x_{k+2})\), that of \((-1)^2\) on the interval \((x_{k+2}, x_{k+3})\), \(\ldots\), and it has the sign of \((-1)^{n-k-1}\) for \(x \gt x_{n-1}\). We therefore have

Replacing the function \(f\) in formula (1.1.20) by the polynomial \(h_k\), we have, by conditions (1.1.28), \(\lambda_0 + \lambda_1 + \ldots + \lambda_k + \lambda_n h_k(x_n) = 0\), and it follows that

But by inequality (1.1.29) and \((-1)^n \lambda_n \gt 0\), the right-hand side of formula (1.1.30) is positive, and therefore Theorem 1.1.2 is proved.

Indeed, by formulas (1.1.8) we have formulas (1.1.13), and since, by inequalities (1.1.27), the right-hand sides of formulas (1.1.13) are different from zero, it follows that the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_n\) are of degree \(n-1\).

Indeed, by the corollary of Theorem 1.1.2, the derivatives \(\varphi_1^{(n-2)}, \varphi_2^{(n-2)}, \ldots, \varphi_n^{(n-2)}\) are polynomials of the first degree, which have at most one root on the respective intervals \((x_0, x_1]\), \((x_1, x_2]\), \(\ldots\), \((x_{n-2}, x_{n-1}]\), \((x_{n-1}, x_n)\).

But we have

and therefore the derivatives \(\varphi_1^{(n-2)}, \varphi_n^{(n-2)}\) do not vanish on the intervals \((x_0, x_1]\), \([x_{n-1}, x_n)\). It then follows that the derivative \(\varphi^{(n-2)}\) has at most \(n-2\) zeros on the interval \((x_1, x_{n-1})\), and therefore on the interval \((x_0, x_n)\). Theorem 1.1.3 is thus proved.

1.1.5. It is very important to study the sign of the function \(\varphi\) on the interval \((x_0, x_n)\). For \(n = 1\), formula (1.1.19) becomes

where \(\varphi(s) = \dfrac{1}{x_1 - x_0}\), and therefore the function \(\varphi\) is positive on the interval \((x_0, x_1)\).

For \(n=2\) we have \(\varphi_1(x) = \dfrac{x-x_0}{(x_1-x_0)(x_2-x_0)}\), \(\varphi_2(x) = \dfrac{x_2-x}{(x_2-x_0)(x_2-x_1)}\), and therefore the function \(\varphi\) is positive on the interval \((x_0, x_2)\).

In general we have the following theorem:

For the proof we assume that \(n \gt 2\) and observe that, by the boundary conditions (1.1.6), the function \(\varphi\) is continuous together with its successive derivatives \(\varphi', \varphi'', \ldots, \varphi^{(n-2)}\) on the interval \([x_0, x_n]\). In addition, we have

Let us first prove that the function \(\varphi\) does not vanish on the interval \((x_0, x_n)\). To this end we assume the contrary, namely that the function \(\varphi\) vanishes at least once on this interval, at a point \(\xi_1^{(0)}\). Applying Rolle's theorem to the function \(\varphi\) and to the intervals \([x_0, \xi_1^{(0)}]\), \([\xi_1^{(0)}, x_n]\), we deduce that the derivative \(\varphi'\) vanishes at least at two points \(\xi_1^{(1)}, \xi_2^{(1)}\) of the interval \((x_0, x_n)\). Then, applying Rolle's theorem to the function \(\varphi'\) and to the intervals \([x_0, \xi_1^{(1)}]\), \([\xi_1^{(1)}, \xi_2^{(1)}]\), \([\xi_2^{(1)}, x_n]\), we deduce that the derivative \(\varphi''\) vanishes at least at three points \(\xi_1^{(2)}, \xi_2^{(2)}, \xi_3^{(2)}\) of the interval \((x_0, x_n)\), \(\ldots\), and so on. We arrive at the conclusion that the derivative \(\varphi^{(n-2)}\) vanishes at least at \(n-1\) points \(\xi_1^{(n-2)}, \xi_2^{(n-2)}, \ldots, \xi_{n-1}^{(n-2)}\) of the interval \((x_0, x_n)\). But this contradicts Theorem 1.1.3, which shows that the number of zeros of \(\varphi^{(n-2)}\) on the interval \((x_0, x_n)\) is at most \(n-2\). We must therefore reject the hypothesis that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_n)\), and consequently it keeps its sign on this interval. We have \(\varphi_1(s) = \lambda_0 \dfrac{(s-x_0)^{n-1}}{(n-1)!} \gt 0\) on the interval \((x_0, x_1]\), and therefore the function \(\varphi\) is positive on the interval \((x_0, x_n)\). Theorem 1.1.4 is thus proved.

To prove it, it suffices to replace the function \(f(x)\) in formula (1.1.19) by \(x^n\) and to observe that its left-hand side becomes equal to 1. Formula (1.1.32) then follows immediately.

Indeed, it suffices to apply the mean value formula to the right-hand side of formula (1.1.19) and to take formula (1.1.32) into account.

§ 1.2. The Lagrange interpolation formula

The representation of the divided difference of a function \(f\) on simple nodes by a definite integral leads in a remarkable way to the Lagrange interpolation formula, to its remainder, and to the Lagrange interpolation polynomial.

1.2.1. Let us consider a function \(f \in C^n[a, b]\) and the simple nodes \(x_1, x_2, \ldots, x_n\) of the interval \((a, b)\), where \(x_1 \lt x_2 \lt \ldots \lt x_n\), as well as a node \(x \in (a, b)\) which, however, does not coincide with any of the nodes \(x_1, x_2, \ldots, x_n\). The node \(x\) may lie to the left of \(x_1\), or to the right of \(x_n\), or in the interval \((x_1, x_n)\), without coinciding with any of the nodes \(x_2, x_3, \ldots, x_{n-1}\).

Let us first assume that \(x \in (x_i, x_{i+1})\), where \(i\) is one of the numbers \(1, 2, \ldots, n-1\). In this case we can consider the integral representation of the divided difference of order \(n\) of the function \(f\) on the nodes \(x_1, \ldots, x_i, x, x_{i+1}, \ldots, x_n\) and write the formula

where we have denoted the function \(\varphi\) of formula (1.1.19) by \(\varphi_i(x, s)\), in order to indicate that it depends on the node \(x \in (x_i, x_{i+1})\).

The left-hand side of formula (1.2.1) can be replaced by

and formula (1.2.1) can be written in the form

Introducing the polynomial

we have \((-1)^n \dfrac{V(x, x_1, \ldots, x_n)}{V(x_1, \ldots, x_n)} = \omega(x)\), and we can write formula (1.2.2) in the form

The first term of formula (1.2.4) is a polynomial \(L_{n-1}(x)\), which we consider on the whole interval \([a, b]\) and whose degree is at most \(n-1\). We shall write

The polynomial \(L_{n-1}(x)\) is related to the function \(f\) by the following two theorems:

Indeed, replacing \(x\) by \(x_k\) in formula (1.2.5), we obtain

and therefore we have \(L_{n-1}(x_k) = f(x_k)\) for \(k = 1, 2, \ldots, n\).

Indeed, the equations which determine the coefficients of the polynomial \(P_{n-1}\) are

and the determinant of this system is different from zero.

Eliminating the constants \(A_{n-1}, A_{n-2}, \ldots, A_0\) between equations (1.2.7) and (1.2.9), we find

and, solving this equation for \(P_{n-1}(x)\), we find that \(P_{n-1}(x) = L_{n-1}(x)\).

Formula (1.2.4) can be written in the form

this appears as an approximation formula, which is called the Lagrange interpolation formula. The function \(f\) is approximated on the interval \([a, b]\) by the polynomial \(L_{n-1}\), which is called the Lagrange interpolation polynomial.

The remainder of formula (1.2.10) is

it depends on \(x\) and on the index which shows that \(x \in (x_i, x_{i+1})\).

1.2.2. If \(x \lt x_1\), we have the integral representation

where we have denoted the function \(\varphi(s)\) of formula (1.1.19) by \(\varphi_0(x, s)\) in order to indicate that it corresponds to the node \(x \lt x_1\).

Proceeding as in subsection 1.2.1, we arrive at the Lagrange interpolation formula

where the remainder of this formula is

If \(x \gt x_n\), we have the integral representation

where we have denoted the function \(\varphi\) of formula (1.1.19) by \(\varphi_n(x, s)\) in order to show that it corresponds to the node \(x \gt x_n\).

Proceeding as in subsection 1.2.1, we arrive at the Lagrange interpolation formula

where the remainder \(R_n(x)\) is given by the formula

1.2.3. The function \(\varphi_i(x, s)\) is positive on the interval \((x_1, x_n)\), \((i = 1, 2, \ldots, n-1)\), as are the functions \(\varphi_0(x, s)\), \(\varphi_n(x, s)\) on the intervals \((x, x_n)\), \((x_1, x)\). We have

and

Hence we obtain for \(R_i(x)\) the well-known estimate

where

and

§ 1.3. The case of nodes in arithmetic progression

In this section we consider the divided difference of order \(n\) of a function \(f\) on the nodes \(x_0, x_1, \ldots, x_n\) in arithmetic progression with common difference \(h = x_1 - x_0\). We shall thus be led to define the difference of order \(n\) of a function \(f\) on the nodes \(x_0, x_1, \ldots, x_n\) in arithmetic progression, and to establish its integral representation when \(f \in C^n[x_0, x_n]\).

1.3.1. As is well known, the divided difference \([x_0, x_1, \ldots, x_n; f]\) has the recurrence property

which is easily proved starting from the fundamental formula (1.1.17). In order to be able to apply formula (1.3.1) also for \(n = 1\), we agree to write \([x; f] = f(x)\).

In order to apply formula (1.3.3) also for \(n = 1\), we agree to write \(\Delta^0 f(x) = f(x)\).

Taking into account the first part of this definition, we can write

We shall prove formula (1.3.5) by the method of complete induction. For \(n = 1\) the formula is true, because it coincides with formula (1.3.4).

Let us assume that formula (1.3.5) is true for \(n\) arbitrary nodes in arithmetic progression, and let us prove that it is true for \(n + 1\) nodes in arithmetic progression.

To this end we apply formula (1.3.1) and obtain

which means, by formula (1.3.3), that

and therefore Theorem 1.3.1 is proved.

Indeed, this formula is true for \(n = 1\), because it coincides with the defining formula (1.3.2). Further, formula (1.3.6′) is proved by the method of complete induction, using the defining formula (1.3.2).

1.3.2. Using the integral representation of the divided difference

and formula (1.3.5), we deduce the following theorem

1.3.3. We shall give some important applications of formula (1.3.7).

1°. Let us first put \(f(x) = (x - x_0)^r\) in formula (1.3.7), where \(r \leq n\). Taking also formula (1.3.9) into account, we shall then have the identities

2°. Putting next \(f(x) = (x - x_0)^{n+1}\) in formula (1.3.7), we shall have

But according to formula (1.1.19), we have

The left-hand side of formula (1.3.12) is

and therefore, by formula (1.3.12), we have

Returning to formula (1.3.11) and taking formula (1.3.8) into account, we shall have the identity

Identities (1.3.10) and (1.3.13) are known. What is interesting is that we have deduced them from the integral representation (1.3.7).

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Română

§ 2.1. The second boundary value problem

In this section we shall study a new boundary value problem which will lead us to the definition of the divided difference of a function \(f\) on the nodes \(x_0, x_1, \ldots, x_k\) of multiplicities \(n_0, n_1, \ldots, n_k\), where \(x_0 \lt x_1 \lt \ldots \lt x_k\) and where \(n + 1 = n_0 + n_1 + \ldots + n_k\), as well as to its representation by a definite integral when \(f \in C^n[x_0, x_k]\).

2.1.1. Let us consider the nodes \(x_0, x_1, \ldots, x_k\), where

which are of multiplicities \(n_0, n_1, \ldots, n_k\), and let us denote by \(n\) the number defined by

To the intervals \([x_0, x_1]\), \([x_1, x_2], \ldots, [x_{k-1}, x_k]\) we attach the polynomials \(\varphi_1, \varphi_2, \ldots \varphi_k\), solutions of the differential equations

and we compute the integrals \(\int_{x_0}^{x_1} \varphi_1 f^{(n)}\, ds\), \(\int_{x_1}^{x_2} \varphi_2 f^{(n)}\, ds, \ldots, \int_{x_{k-1}}^{x_k} \varphi_k f^{(n)}\, ds\).

Applying the generalized integration by parts formula, we shall have

Adding formulas (2.1.4) term by term, we obtain a formula of the form

where the function \(\varphi\) coincides with the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_k\) on the intervals \([x_0, x_1]\), \([x_1, x_2], \ldots, [x_{k-1}, x_k]\), and where \(\mathscr{P}[f]\) depends on the values of \(f, f', \ldots, f^{(n-1)}\) at the nodes \(x_0, x_1, \ldots, x_k\).

In connection with formula (2.1.5) we pose the following

For this we must introduce the boundary conditions

and also

The solution of the differential equations (2.1.3) which satisfies the boundary conditions (2.1.7) relative to the nodes \(x_0, x_1, \ldots, x_{k-1}\) is given by the formulas

where the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_{k-1}, \lambda_{k-1}^{(1)}, \ldots, \lambda_{k-1}^{(n_{k-1}-1)}\) are constants. The number of these constants is \(n_0 + n_1 + \ldots + n_{k-1} = n - n_k + 1\).

The constants \(\lambda_0, \ldots, \lambda_{k-1}^{(n_{k-1}-1)}\) are determined by writing that the boundary conditions at the point \(x_k\) are satisfied. We then have the equations

We adopt the convention that in equations (2.1.10) the terms in which the exponents of \(x_k - x_0, x_k - x_1, \ldots, x_k - x_{k-1}\) are negative are to be taken as zero.

Equations (2.1.10) can also be written in the form

The matrix of the coefficients of this system is

The entries of the 2^nd, 3^rd, \(\ldots\), \((n_0)\)^th columns are the derivatives of the first, the 2^nd, \(\ldots\), the \((n_0 - 1)\)^th order of the entries of the first column. In the matrix there is a second group of columns relative to the column of \((x_k - x_1)^{n_k}\), a third one relative to the column of \((x_k - x_2)^{n_k}\), \(\ldots\), and a last group relative to the column of \((x_k - x_{k-1})^{n_k}\).

Matrix (2.1.12) has rank \(n - n_k\).

Indeed, it is easily shown that the determinant obtained from matrix (2.1.12) by deleting the last column is not zero, because it differs by a nonzero factor from the generalized Vandermonde determinant

which is not zero.

Thus the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_{k-1}, \lambda_{k-1}^{(1)}, \ldots, \lambda_{k-1}^{(n_{k-1}-1)}\) are determined up to a factor. It follows that the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_k\) given by formulas (2.1.9) are determined up to a factor, and we can use them.

In formula (2.1.5) we can replace the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_k\) by the solution of the boundary value problem. Then \(\mathscr{P}[f]\) reduces to

But by formulas (2.1.9) we have

We shall also write

Since the function \(\varphi_k\) is determined by the last formula (2.1.9) by means of the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_{k-1}, \lambda_{k-1}^{(1)}, \ldots, \lambda_{k-1}^{(n_{k-1}-1)}\), the new coefficients \(\lambda_k, \lambda_k^{(1)} \ldots\) introduced by formulas (2.1.15) are determined, like the other coefficients \(\lambda_0, \ldots, \lambda_{k-1}^{(n_{k-1}-1)}\), up to a factor.

Taking all these formulas into account, formula (2.1.13) becomes

Replacing in formula (2.1.5) the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_k\) by the solution of the boundary value problem, we can write

2.1.2. An important consequence of this formula is that the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_k, \lambda_k^{(1)}, \ldots, \lambda_k^{(n_k-1)}\) satisfy the equations which we would deduce from formula (2.1.17) by replacing the function \(f\) successively by \(1, x, \ldots, x^{n-1}\). These equations are the following:

The matrix of system (2.1.18) is

Let us denote by \(D_0, D_0^{(1)}, \ldots, D_0^{(n_0-1)}, \ldots, D_k, D_k^{(1)}, \ldots, D_k^{(n_k-1)}\) the determinants of the matrices deduced from matrix (2.1.19) by deleting successively the 1^st, the 2^nd, \(\ldots\), the last column. Then the solution of system (2.1.18) is given by the formulas

where \(H\) is a constant which we shall determine.

Replacing in formula (2.1.16) the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_k, \lambda_k^{(1)}, \ldots, \lambda_k^{(n_k-1)}\) by the values deduced from formulas (2.1.20), we obtain

But the large bracket multiplied by \((-1)^n\) is equal to the determinant

and therefore

According to the last boundary condition (2.1.8), \(\mathscr{P}[f]\) must reduce to 1 when \(f(x)\) is replaced by \(x^n\). Formula (2.1.23) shows that we must take

and therefore the expression of \(\mathscr{P}[f]\) is

All the constants \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}, \ldots, \lambda_k, \lambda_k^{(1)}, \ldots, \lambda_k^{(n_k-1)}\) are now determined, and the boundary value problem is completely solved.

Formula (2.1.25) shows that

where \(\Delta[f]\) is the determinant (2.1.22).

Formula (2.1.27) leads to the following theorem:

The integral representation (2.1.28) can also be written in the following form:

in which we have displayed the coefficients of \(f(x_0), f'(x_0), \ldots, f^{(n_0-1)}(x_0), \ldots, f(x_k), f'(x_k) \ldots, f^{(n_k-1)}(x_k)\).

§ 2.2. On a property of the coefficients \(\lambda_0, \lambda_1, \ldots, \lambda_{k-1}\) in formula (2.1.29)

In this section we shall prove that the coefficients \(\lambda_0, \lambda_1, \ldots, \lambda_j\) in formula (2.1.29) have the property expressed by \(\lambda_0 + \lambda_1 + \ldots + \lambda_j \neq 0\), for \(j = 0, 1, \ldots, k - 1\).

Indeed, by formulas (2.1.20) we have \(\lambda_k^{(n_k-1)} = (-1)^n H D_k^{(n_k-1)}\).

But the determinant \(D_k^{(n_k-1)}\) is the generalized Vandermonde determinant

and, taking formula (2.1.24) into account, we have

which means that \(\lambda_k^{(n_k-1)} \neq 0\), and therefore the theorem is proved.

Theorem 2.2.1 is also valid in the case when the node \(x_k\) is simple, that is, in the case when \(n_k = 1\).

In this case formula (2.2.1) is replaced by

and therefore \(\lambda_k \neq 0\).

To prove this theorem, let us introduce the interpolation polynomial \(h_j\) defined by the following conditions:

and

where it is assumed that \(n_k \gt 1\).

The number of conditions which determine the polynomial \(h_j\) is \(n_0 + n_1 + \ldots + n_{k-1} + n_k - 1 = n\), whence it follows that the degree of the polynomial \(h_j\) is at most \(n-1\).

From conditions (2.2.3) and (2.2.4) one can form the system of equations which determines the coefficients of the polynomial \(h_j(x) = A_0 x^{n-1} + A_1 x^{n-2} + \ldots + A_{n-1}\).

The determinant of the system in \(A_0, A_1, \ldots, A_{n-1}\) is the generalized Vandermonde determinant

which is different from zero, so that the polynomial \(h_j\) is uniquely determined.

It may be added that the polynomial \(h_j\) is not identically zero, since we have \(h_j(x_0) = 1\), whereas \(h_j(x_k) = 0\), and this holds for \(j = 0, 1, \ldots, k-1\).

Replacing the function \(f\) in formula (2.1.29) by the polynomial \(h_j\), we obtain the formula

In this formula we have \(\lambda_k^{(n_k-1)} \neq 0\), by Theorem 2.2.1, and in order to prove Theorem 2.2.2 it remains to show that \(h_j^{(n_k-1)}(x_k) \neq 0\).

Let us denote by \(p\) the highest order of the derivatives occurring in conditions (2.2.3) and (2.2.4). Let us also denote by \(m_1\) the number of nodes \(x_0, x_1, \ldots, x_k\) at which \(h_j'\) vanishes, then by \(m_2\) the number of nodes \(x_0, x_1, \ldots, x_k\) at which \(h_j''\) vanishes, …, and finally by \(m_p\) the number of nodes \(x_0, x_1, \ldots, x_k\) at which \(h_j^{(p)}\) vanishes. Expressing the number of conditions (2.2.3) and (2.2.4) in two different ways, we obtain the equality

whence it follows that

Applying Rolle's theorem successively, it follows that the derivative \(h_j'\) has \(k + m_1 - 1\) distinct roots on the interval \([x_0, x_k]\), that the derivative \(h_j''\) has \(k + m_1 + m_2 - 2\) distinct roots on the interval \([x_0, x_k]\), …, and that the derivative \(h_j^{(p)}\) has

roots on the interval \([x_0, x_k]\).

The degree of the polynomial \(h_j\) being at most \(n-1\), that of \(h_j^{(p)}\) will be at most \(n-p-1\). But we have shown above that \(h_j^{(p)}\) has \(n-p-1\) distinct roots on the interval \([x_0, x_k]\).

Hence the degree of the polynomial \(h_j^{(p)}\) is \(n-p-1\), or less than \(n-p-1\). Let us prove that it is impossible for the degree of \(h_j^{(p)}\) to be less than \(n-p-1\).

Indeed, let us suppose that the polynomial \(h_j^{(p)}\) had degree less than \(n-p-1\). Since this polynomial has \(n-p-1\) distinct roots, it would be identically zero. It follows that the derivative \(h_j^{(p-1)}\) must be a constant, and this constant is zero, because the derivative vanishes at one of the points \(x_0, x_1, \ldots, x_j\). Hence \(h_j^{(p-1)}\) is an identically zero polynomial. Continuing this reasoning, we reach the conclusion that the polynomial \(h_j\) must be a constant, which is impossible, since \(h_j(x_0) = 1\), whereas \(h_j(x_k) = 0\).

Thus the polynomial \(h_j^{(p)}\) is of degree \(n-p-1\) and has \(n-p-1\) distinct roots on the interval \([x_0, x_k]\).

This being established, we can complete the proof of Theorem 2.2.3 and prove that \(h_j^{(n_k-1)}(x_k) \neq 0\).

Let us first treat the case \(n_k - 2 \lt p\) and suppose that \(h_j^{(n_k-1)}(x_k) = 0\). Repeating the reasoning based on Rolle's theorem, we are led to assert that the derivative \(h_j^{(p)}\) must have \(n-p\) distinct roots on the interval \([x_0, x_k]\), which is false, because we have proved that the number of roots of \(h_j^{(p)}\) is \(n-p-1\).

Let us finally treat the case \(n_k - 2 = p\) and suppose that \(h_j^{(n_k-1)}(x_k) = h_j^{(p+1)}(x_k) = 0\). Since the derivative \(h_j^{(p)}\) has \(n-p-1\) distinct roots on the interval \([x_0, x_k]\), its derivative \(h_j^{(p+1)}\) has \(n-p-2\) distinct roots on the interval \((x_0, x_k)\) and, in addition, the root \(x_k\), by our hypothesis. In all, the derivative \(h_j^{(p+1)}\) has \(n-p-1\) distinct roots on the interval \((x_0, x_k)\). But this is impossible, because \(h_j^{(p+1)}\), being a polynomial of degree \(n-p-2\), has at most \(n-p-2\) roots on the interval \([x_0, x_k]\). We have thus reached a contradiction, and we must drop the hypothesis that \(h_j^{(n_k-1)}(x_k) = 0\), which means that \(h_j^{(n_k-1)}(x_k) \neq 0\).

2.2.4. We can now prove Theorem 2.2.2 in the case \(n_k \gt 1\).

Indeed, we have formula (2.2.5), and we have proved Theorems 2.2.1 and 2.2.3, which show that \(\lambda_k^{(n_k-1)} \neq 0\) and \(h_j^{(n_k-1)}(x_k) \neq 0\). It follows that \(\lambda_0 + \lambda_1 + \ldots + \lambda_j \neq 0\) for \(j = 0, 1, \ldots, k-1\).

It remains to prove Theorem 2.2.2 in the case \(n_k = 1\).

2.2.5. To this end we introduce the interpolation polynomial, which we again denote by \(h_j\), defined by the following conditions:

and

where \(j = 0, 1, \ldots, k-2\).

The number of conditions which determine the polynomial \(h_j\) is

whence it follows that the polynomial \(h_j\) is of degree at most \(n-1\).

As in subsection 2.2.2, one easily sees that the polynomial \(h_j\) is uniquely determined. Moreover, \(h_j\) is not identically zero, because, whatever \(j = 0, 1, \ldots, k-2\) may be, we have \(h_j(x_0) = 1\), whereas \(h_j(x_{k-1}) = 0\).

Replacing the function \(f\) in formula (2.1.29) by the polynomial \(h_j\), we have

By Theorem 2.2.1, formula (2.2.1′), we have \(\lambda_k \neq 0\).

It remains to prove that \(h_j(x_k) \neq 0\), which is done by the following theorem:

As in subsection 2.2.3, let us denote by \(p\) the highest order of the derivatives occurring in conditions (2.2.7) and (2.2.8). We also denote by \(m_1\) the number of points \(x_0, x_1, \ldots, x_{k-1}\) at which \(h_j'\) vanishes, by \(m_2\) the number of points \(x_0, x_1, \ldots, x_{k-1}\) at which \(h_j''\) vanishes, …, and by \(m_p\) the number of points \(x_0, x_1, \ldots, x_{k-1}\) at which \(h_j^{(p)}\) vanishes. We have

Applying Rolle's theorem successively, we deduce that \(h_j'\) has \(k + m_1 - 2\) roots on the interval \([x_0, x_{k-1}]\), then that \(h_j''\) has \(k + m_1 + m_2 - 3\) roots on the interval \([x_0, x_{k-1}]\), …, and that \(h_j^{(p)}\) has \(N = k + m_1 + m_2 + \ldots + m_p - (p + 1) = n - p - 1\) roots on the interval \([x_0, x_{k-1}]\).

As in subsection 2.2.3, one deduces that the derivative \(h_j^{(p)}\) is a polynomial of degree \(n-p-1\) which vanishes at \(n-p-1\) distinct points of the interval \([x_0, x_{k-1}]\).

This being established, one easily proves that \(h_j(x_k) \neq 0\).

Indeed, let us suppose that \(h_j(x_k)\) were zero. Then, applying Rolle's theorem successively to \(h_j\), we reach the conclusion that \(h_j^{(p)}\) must have one more root, to the right of the last root exhibited above. But this is impossible, because it has been proved that \(h_j^{(p)}\) has only \(n-p-1\) roots on the interval \([x_0, x_{k-1}]\).

Hence \(h_j(x_k) \neq 0\), and therefore Theorem 2.2.4 is proved.

Indeed, the coefficients of \(s^{n-1}\) in these polynomials are

and, by Theorem 2.2.2, they are all different from zero.

§ 2.3. On a property of the coefficients \(\lambda_0, \lambda_0^{(1)}, \ldots, \lambda_0^{(n_0-1)}\) and \(\lambda_k, \lambda_k^{(1)}, \ldots, \lambda_k^{(n_k-1)}\) of formula (2.1.29)

The aim of this section is to prove a theorem on the coefficients of \(f(x_0), f'(x_0), \ldots, f^{(n_0-1)}(x_0)\) and \(f(x_k), f'(x_k), \ldots, f^{(n_k-1)}(x_k)\) in the divided difference

when the nodes \(x_0, x_1, \ldots, x_k\) have multiplicities \(n_0, n_1, \ldots, n_k\) and \(x_0 \lt x_1 \lt \ldots \lt x_k\). We shall make use of the formula

where \(\Delta[f]\) is the determinant

Clearly it suffices to prove only one of the parts of the theorem. The other part is proved in an analogous way.

We have already proved, in Theorem 2.2.1, that the coefficient \(\lambda_k^{(n_k-1)}\) is different from zero. It remains to prove Theorem 2.3.1 for the coefficients \(\lambda_k, \lambda_k^{(1)}, \ldots, \lambda_k^{(n_k-2)}\).

2.3.2. Let us first prove that \(\lambda_k \neq 0\). This amounts to proving that the determinant

where \(n + 1 = n_0 + n_1 + \ldots + n_{k-1} + n_k\), is different from zero.

For brevity we write the determinant (2.3.2) in the form

We shall put

Then formula (2.3.3) becomes

2.3.3. Let us first prove that the determinant

is different from zero.

To this end we consider the polynomial

which is of degree \(N\), because the coefficient of \(x^N\) is the generalized Vandermonde determinant

which is different from zero. We can write

The polynomial \(\Delta_1\) has all its roots real and lying in the interval \([x_0, x_{k-1}]\). The derivative \(\Delta_1'\) likewise has all its roots real and lying in the same interval. It follows that \(\Delta_1'(x) \neq 0\) for \(x \gt x_{k-1}\), and in particular that \(\Delta_1'(x_k) \neq 0\). But \(\Delta_1'(x_k) = d_1\), and therefore we have proved that \(d_1 \neq 0\).

2.3.4. To prove that the determinant (2.3.5) is different from zero, we use the method of complete induction.

Let us suppose that the determinant (2.3.5) is different from zero for the index \(j\), which was proved in subsection 2.3.3 for \(j = 1\), and let us prove that the determinant

is different from zero.

To this end we introduce the polynomial

which is of degree \(N + j\), because the coefficient of \(x^{N+j}\) is the determinant \(d_j\), which we have assumed to be different from zero. We can write

where \(U_j(x)\) is a polynomial of degree \(j\).

The polynomial \(\Delta_{j+1}\) has \(n_0 + n_1 + \ldots + n_{k-1} = N\) real roots lying in the interval \([x_0, x_{k-1}]\). The derivative \(\Delta_{j+1}'\) has \(N-1\) real roots lying in the same interval, and the root \(x_k\) of multiplicity \(j\), which means that \(\Delta_{j+1}'\) has all its roots real and lying in the interval \([x_0, x_k]\). It then follows that the derivative \((\Delta_{j+1}')^{(j)} = \Delta_{j+1}^{(j+1)}\) has all its roots real and lying in the interval \([x_0, x_k)\). We deduce that \(\Delta_{j+1}^{(j+1)}(x_k) \neq 0\). But \(\Delta_{j+1}^{(j+1)}(x_k) = d_{j+1}\), and therefore \(d_{j+1} \neq 0\).

Thus we have proved that the coefficient \(\lambda_k\) is different from zero.

2.3.5. To prove that \(\lambda_k^{(l)} \neq 0\), where \(l\) is one of the numbers \(1, 2, \ldots, n_k-2\), we must prove that the determinant

where \(n + 1 = n_0 + n_1 + \ldots + n_{k-1} + n_k\), is different from zero.

For brevity, let us write the preceding determinant in the form

We shall put \(n_0 + n_1 + \ldots + n_{k-1} = N\), \(n_k - 1 = l + j\), and therefore \(n = N + l + j\). Then the determinant (2.3.11) becomes

Let us first prove that for \(j = 1\) the determinant

is different from zero.

To this end we consider the polynomial

which is of degree \(N + l\), because the coefficient of \(x^{N+l}\) is the generalized Vandermonde determinant

which is not zero.

We can write

All the roots of the polynomial \(\Delta_{l+1}\) are real and lie in the interval \([x_0, x_k]\). It follows that the roots of the successive derivatives of this polynomial are real and lie in the same interval. In particular, the roots of \(\Delta_{l+1}^{(l+1)}\) are all contained in the interval \([x_0, x_k)\), so that \(\Delta_{l+1}^{(l+1)}(x_k) \neq 0\). But \(\Delta_{l+1}^{(l+1)}(x_k) = d_{l+1}^{(l)}\), and therefore we have proved that \(d_{l+1}^{(l)} \neq 0\).

2.3.6. We shall prove by the method of complete induction that the determinant \(d_{l+j}^{(l)}\) is different from zero. We have already seen in subsection 2.3.5 that \(d_{l+1}^{(l)} \neq 0\). We shall suppose that for the index \(j\) we have \(d_{l+j}^{(l)} \neq 0\), and we shall prove that the determinant

is also different from zero.

To this end we consider the polynomial

which is of degree \(N + l + j\), because the coefficient of \(x^{N+l+j}\) is the determinant \(d_{l+j}^{(l)}\), which we have assumed to be different from zero.

We can write the polynomial \(\Delta_{l+j+1}\) in the form

where \(U_j(x)\) is a polynomial of degree \(j\).

The polynomial \(\Delta_{l+j+1}\), of degree \(N + l + j\), has \(N + l\) roots on the interval \([x_0, x_k]\). It follows that the derivative \(\Delta_{l+j+1}^{(l)}\) has \(N + j\) roots on the interval \([x_0, x_k]\). The polynomial \(\Delta_{l+j+1}^{(l)}\) no longer has the root \(x_k\), because \(\Delta_{l+j+1}^{(l)}(x_k)\) is a generalized Vandermonde determinant, which is not zero. But the derivative \((\Delta_{l+j+1}^{(l)})' = \Delta_{l+j+1}^{(l+1)}\) has the root \(x_k\) of multiplicity \(j\). Finally, the derivative \((\Delta_{l+j+1}^{(l+1)})^{(j)} = \Delta_{l+j+1}^{(l+j+1)}\), of degree \(N - 1\), has all its \(N-1\) roots real and lying in the interval \([x_0, x_k]\). We therefore have \(\Delta_{l+j+1}^{(l+j+1)}(x_k) \neq 0\). But \(\Delta_{l+j+1}^{(l+j+1)}(x_k) = d_{l+j+1}^{(l)}\), and therefore \(d_{l+j+1}^{(l)} \neq 0\).

Thus Theorem 2.3.1 is proved.

§ 2.4. The fundamental theorem on the function \(\varphi\)

The aim of this section is to prove that the function \(\varphi\) in the integral representation

is positive on the interval \((x_0, x_k)\), as in the case of simple nodes (Theorem 1.1.4).

2.4.1. Let us denote by \(n-n_i-1\) the smallest number of the sequence

and let us observe that the function \(\varphi\) is continuous on the interval \([x_0, x_k]\), together with its successive derivatives up to order \(n-n_i-1\). The derivatives of order higher than \(n-n_i-1\) are continuous on certain subintervals of \([x_0, x_k]\). In particular, the function \(\varphi\) is continuous, together with its successive derivatives up to the \(n^{\text{th}}\) order, on each interval \([x_0, x_1], [x_1, x_2], \ldots, [x_{k-1}, x_k]\).

The theorem on the sign of the function \(\varphi\) on the interval \((x_0, x_k)\) will be easily deduced from the following theorem:

We shall prove this theorem in stages, using several auxiliary lemmas.

2.4.2. Let us consider an interval \([x_j, x_{j+1})\) with \(j \gt 0\), and let us attach the numbers \(n-n_j-1\) and \(n-n_{j+1}-1\) to its endpoints. We have

It suffices to prove the first part of the lemma.

To this end we shall prove that the derivative \(\varphi^{(n-n_j-1)}\) cannot have \(n_j + 1\) zeros on the interval \([x_j, x_{j+1})\).

Indeed, let us suppose that on the interval \([x_j, x_{j+1})\) the number of zeros of \(\varphi^{(n-n_j-1)}\) is equal to \(n_j + 1\). Since this number is at least 2, we may apply Rolle's theorem. We deduce that the derivative \(\varphi^{(n-n_j)}\) has \(n_j\) zeros on the interval \([x_j, x_{j+1})\), that \(\varphi^{(n-n_j+1)}\) has \(n_j - 1\) zeros on the same interval, \(\ldots\), and that \(\varphi^{(n-1)}\) must have one zero on the same interval. But this is impossible because, by Theorem 2.2.5, the derivative \(\varphi^{(n-1)}\) is equal to the constant \(\lambda_0 + \lambda_1 + \ldots + \lambda_j\), which is different from zero. Hence the number of zeros of the derivative \(\varphi^{(n-n_j-1)}\) on the interval \([x_j, x_{j+1})\) is at most \(m_j = n_j\).

The second part of the lemma is proved in the same way.

2.4.3. Let us consider three consecutive nodes \(x'_1, x'_2, x'_3\) from the set of nodes \(x_1, x_2, \ldots, x_{k-1}\), with their multiplicities \(n'_1, n'_2, n'_3\).

This lemma is evident if \(n'_2 = n'_1\), because in this case the number of zeros of \(\varphi^{(n-n'_1-1)}\) on the interval \([x'_1, x'_3)\) is equal to the number of zeros of \(\varphi^{(n-n'_2-1)}\) on the same interval, that is, on the interval \([x'_1, x'_2)\) and on \([x'_2, x'_3)\). But by the first lemma this number is \(n'_2 + n'_2 = 2n'_2\), and therefore the lemma is proved.

Let us now suppose that \(n-n'_1-1 \lt n-n'_2-1\) and let us prove that the number of zeros of the derivative \(\varphi^{(n-n'_1-1)}\) on the interval \([x'_1, x'_3)\) is \(m = n'_1 + n'_2\). To this end we first observe that the derivative \(\varphi^{(n-n'_1-1)}\) is continuous on the interval \([x'_1, x'_3)\), together with its successive derivatives up to order \(n-n'_2-1\). Let us prove that it is impossible for \(\varphi^{(n-n'_1-1)}\) to have \(n'_1 + n'_2 + 1\) zeros on the interval \([x'_1, x'_3)\).

Indeed, supposing that \(\varphi^{(n-n'_1-1)}\) had \(n'_1 + n'_2 + 1\) zeros on the interval \([x'_1, x'_3)\), the derivative \(\varphi^{(n-n'_1)}\) would have to have \(n'_1 + n'_2\) zeros on the interval \([x'_1, x'_3), \ldots\), and the derivative \(\varphi^{(n-n'_2-1)}\) would have to have \(2n'_2 + 1\) zeros on the interval \([x'_1, x'_3)\). But this is impossible because, by the first lemma, \(\varphi^{(n-n'_2-1)}\) has at most \(n'_2\) zeros on the interval \([x'_1, x'_2)\) and \(n'_2\) zeros on the interval \([x'_2, x'_3)\), that is, at most \(2n'_2\) zeros on the interval \([x'_1, x'_3)\). Comparing these two results we find a contradiction, and therefore the number of zeros of \(\varphi^{(n-n'_1-1)}\) on the interval \([x'_1, x'_3)\) is at most \(m = n'_1 + n'_2\).

The second part of the lemma is proved analogously.

2.4.4. Let us consider a group of consecutive nodes \(x'_1, x'_2, \ldots, x'_r\) from the set \(x_1, x_2, \ldots, x_k\), where \(r \gt 3\), with their multiplicities, which we denote by \(n'_1, n'_2, \ldots, n'_r\). We denote by \(n-n'-1\) the smallest number of the sequence

We have

For \(r = 3\) the sequence (2.4.6) reduces to its first term and the third lemma coincides with the second lemma; hence it is true.

To prove the third lemma we use the method of complete induction, and we already know that this lemma is true for \(r = 3\), because in this case the third lemma coincides with the second lemma. We shall suppose that the third lemma has been proved for \(r \lt l\) and we shall prove it for \(r = l\).

For this purpose we denote by \(n - n'_j - 1\) the smallest number of the sequence

and by \(x'_j\) one of the points \(x'_2, x'_3, \ldots, x'_{l-1}\) which corresponds to \(n - n'_j - 1\). We obviously have \(j \lt l\).

If \(n - n'_1 - 1 = n - n'_j - 1\), that is, \(n'_1 = n'_j\), then the number of zeros of \(\varphi^{(n-n'_1-1)}\) on the interval \([x'_1, x'_l)\) is equal to the number of zeros of \(\varphi^{(n-n'_j-1)}\) on the same interval. Since we have assumed the third lemma for \(r = j \lt l\), we may apply it, and hence the number of zeros of \(\varphi^{(n-n'_j-1)}\) on the interval \([x'_1, x'_l)\) is equal to the sum of the numbers of zeros of \(\varphi^{(n-n'_j-1)}\) on the intervals \([x'_1, x'_j)\) and \([x'_j, x'_l)\); it is at most equal to \((n'_2 + n'_3 + \ldots + n'_j) + (n'_j + n'_{j+1} + \ldots + n'_{l-1})\), and since \(n'_j = n'_1\), the preceding number is \(m = n'_1 + n'_2 + \ldots + n'_{l-1}\), which proves the third lemma.

Let us now suppose that \(n-n'_1-1 \lt n-n'_j-1\) and prove that the derivative \(\varphi^{(n-n'_1-1)}\) cannot have more than \(m = n'_1 + n'_2 + \ldots + n'_{l-1}\) zeros on the interval \([x'_1, x'_l)\).

To this end let us assume the contrary, namely that \(\varphi^{(n-n'_1-1)}\) had

zeros on the interval \([x'_1, x'_l)\). By the boundary conditions (2.1.7) and the hypothesis that \(n-n'_j-1\) is the smallest of the numbers of the sequence (2.4.9), the function \(\varphi\) is continuous on the interval \([x'_1, x'_l)\), together with its successive derivatives up to order \(n - n'_j - 1\). Then, applying Rolle's theorem, we deduce that the derivative \(\varphi^{(n-n'_1)}\) would have to have \(n'_1 + n'_2 + \ldots + n'_{l-1}\) zeros on the interval \([x'_1, x'_l), \ldots\), and that the derivative \(\varphi^{(n-n'_1+n'_1-n'_j-1)} = \varphi^{(n-n'_j-1)}\) would have to have

zeros on the interval \([x'_1, x'_l)\).

By the third lemma, which we have assumed for \(j \lt l\), the derivative \(\varphi^{(n-n'_j-1)}\) has \(n'_2 + \ldots + n'_j\) zeros on the interval \([x'_1, x'_j)\) and \(n'_j + n'_{j+1} + \ldots + n'_{l-1}\) zeros on the interval \([x'_j, x'_l)\), that is,

zeros on the interval \([x'_1, x'_l)\). There is a contradiction between the numbers (2.4.11) and (2.4.12). We must therefore discard the hypothesis that the derivative \(\varphi^{(n-n'_1-1)}\) had \(n'_1 + n'_2 + \ldots + n'_{l-1} + 1\) zeros on the interval \([x'_1, x'_l)\), and consequently the number of zeros of \(\varphi^{(n-n'_1-1)}\) on the interval \([x'_1, x'_l)\) is at most \(m = n'_1 + n'_2 + \ldots + n'_{l-1}\).

Thus the first part of the third lemma is proved, and the second part is proved analogously.

Let us first prove the first part of the lemma.

To this end we shall prove that it is impossible for the derivative \(\varphi^{(n-n_1-1)}\) to have \(n_1 + 1\) zeros on the interval \((x_0, x_1)\).

Indeed, let us suppose that \(\varphi^{(n-n_1-1)}\) had \(n_1 + 1\) zeros on the interval \((x_0, x_1)\). Then, applying Rolle's theorem, we deduce that \(\varphi^{(n-n_1)}\) would have to have \(n_1\) zeros on the interval \((x_0, x_1), \ldots\) and that \(\varphi^{(n-1)}\) would have to have one zero on the interval \((x_0, x_1)\). But this is impossible by Theorems 2.2.2 and 2.2.5.

The contradiction brought to light by this reasoning shows that the derivative \(\varphi^{(n-n_1-1)}\) has at most \(m = n_1\) zeros on the interval \((x_0, x_1)\).

Let us now pass to the second part of the lemma, supposing that \(n-n_0-1 \geq n-n_1-1\). In this case the reasoning must take into account the boundary conditions at the point \(x_0\)

In this case we have to prove that the derivative \(\varphi^{(n-n_1-1)}\) has at most \(n_0 - 1\) zeros on the interval \((x_0, x_1)\).

To this end let us prove that it is impossible for the derivative \(\varphi^{(n-n_1-1)}\) to have \(n_0\) zeros on the interval \((x_0, x_1)\).

Rolle's theorem can then be applied, taking into account the boundary conditions (2.4.15):

The derivative \(\varphi^{(n-n_1)}\) would have \(n_0\) zeros on the interval \((x_0, x_1)\); \(\varphi^{(n-n_1+1)}\) would likewise have \(n_0\) zeros on the interval \((x_0, x_1), \ldots\), \(\varphi^{(n-n_0)}\) would still have \(n_0\) zeros on the interval \((x_0, x_1)\). Further, since the successive derivatives no longer vanish at \(x_0\), the derivative \(\varphi^{(n-n_0+1)}\) would have \(n_0 - 1\) zeros on the interval \((x_0, x_1), \ldots\), and the derivative \(\varphi^{(n-1)}\) would have one zero on the interval \((x_0, x_1)\). But this is impossible by Theorems 2.2.2 and 2.2.5.

Again we have a contradiction, and therefore the number of zeros of \(\varphi^{(n-n_1-1)}\), when \(n-n_0-1 \geq n-n_1-1\), on the interval \((x_0, x_1)\) is at most \(m = n_0 - 1\).

The fourth lemma is thus proved.

2.4.6. Let us suppose that in the sequence

which corresponds to the nodes \(x_1, x_2, \ldots, x_h\), the last term is smaller than the others.

Then we have:

This lemma is true for \(h = 1\), because in this case it coincides with the fourth lemma.

We shall carry out the proof of this lemma by the method of complete induction. Let us suppose that the fifth lemma has been proved for \(h \lt j\) and let us prove it for \(h = j\).

Let us consider the sequence

and suppose that \(n - n_j - 1\) is the smallest term of the sequence.

Let us further consider the sequence

and suppose that \(n-n_l-1\) is the smallest term of this sequence.

Let us denote by \(x_l\) the node closest to \(x_0\) which corresponds to \(n-n_l-1\).

By the boundary conditions (2.1.7), the function \(\varphi\) is continuous on the interval \([x_0, x_j]\), together with its successive derivatives up to order \(n-n_l-1\).

We shall examine successively the following three cases:

1° Let us first suppose that \(n-n_0-1 \lt n-n_j-1\). Let us prove that the derivative \(\varphi^{(n-n_j-1)}\) has at most \(n_1 + n_2 + \ldots + n_j\) zeros on the interval \((x_0, x_j)\).

To this end let us suppose the contrary, namely that \(\varphi^{(n-n_j-1)}\) had

zeros on the interval \((x_0, x_j)\), and let us show that this hypothesis leads to a contradiction.

We can apply Rolle's theorem to the derivative \(\varphi^{(n-n_j-1)}\) and to its successive derivatives with respect to the interval \([x_0, x_j]\). We deduce that \(\varphi^{(n-n_j)}\) would have at most \(n_1 + n_2 + \ldots + n_j\) zeros on the interval \((x_0, x_j), \ldots\), and that \(\varphi^{(n-n_l-1)}\) would have

zeros on the interval \((x_0, x_j)\).

But we have \(l \lt j\) and \(n-n_0-1 \lt n-n_l-1\), so that the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_l)\) is at most \(n_1 + n_2 + \ldots + n_l\), by the fifth lemma, which we have assumed for \(l \lt j\). On the other hand, the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \([x_l, x_j)\) is given by the third lemma, and it is at most equal to \(n_l + n_{l+1} + \ldots + n_{j-1}\). Hence the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_j)\) is at most equal to

Since there is a disagreement between the numbers (2.4.23) and (2.4.24), we must reject the hypothesis that \(\varphi^{(n-n_j-1)}\) had \(n_1 + n_2 + \ldots + n_j + 1\) zeros on the interval \((x_0, x_j)\), and therefore \(\varphi^{(n-n_j-1)}\) has at most \(n_1 + n_2 + \ldots + n_j\) zeros on the interval \((x_0, x_j)\).

Thus the fifth lemma is proved in the case \(n-n_0-1 \lt n-n_j-1\).

2° Let us suppose that \(n-n_0-1 = n-n_j-1\) and let us prove that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is at most \(n_0 - 1 + n_1 + \ldots + n_{j-1}\).

Let us suppose the contrary, namely that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}\). Let us show that this hypothesis leads to a contradiction.

Because of the boundary conditions (2.1.7) at the point \(x_0\), the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \([x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1} + 1\), since, as \(n_j = n_0\), it follows that \(\varphi^{(n-n_j-1)}(x_0) = \varphi^{(n-n_0-1)}(x_0) = 0\). Applying Rolle's theorem we deduce that \(\varphi^{(n-n_j)}\) has \(n_1 + n_2 + \ldots + n_j\) zeros on the interval \((x_0, x_j), \ldots\), and that \(\varphi^{(n-n_l-1)}\) has

zeros on the interval \((x_0, x_j)\).

But the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_l)\) is, by the fifth lemma, which we have assumed for \(l \lt j\), equal to \(n_1 + n_2 + \ldots + n_l\), and by the third lemma the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \([x_l, x_j)\) is at most \(n_l + n_{l+1} + \ldots + n_{j-1}\). Hence the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \([x_0, x_j)\) is the sum \((n_1 + n_2 + \ldots + n_l) + (n_l + n_{l+1} + \ldots + n_{j-1})\), that is,

Since there is a contradiction between the numbers (2.4.25) and (2.4.26), it follows that we must reject the hypothesis that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}\), and therefore it is at most equal to \(n_0 - 1 + n_1 + \ldots + n_{j-1}\).

The fifth lemma is therefore proved in the case \(n-n_0-1 = n-n_j-1\).

3°. Let us finally suppose that \(n-n_0-1 \gt n-n_j-1\). In this case we shall examine three cases separately: \(n - n_0 - 1 \lt n - n_l - 1\); \(n - n_0 - 1 = n - n_l - 1\); \(n - n_0 - 1 \gt n - n_l - 1\).

Let us prove that in these cases the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is at most \(n_0 - 1 + n_1 + \ldots + n_{j-1}\).

a) Let us first suppose that \(n-n_0-1 \lt n-n_l-1\). Since \(n-n_j-1 \lt n-n_0-1\), we must take into account that, by the boundary conditions (2.1.7) at the point \(x_0\), we have

Let us prove that it is impossible for the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) to be \(n_0 + n_1 + \ldots + n_{j-1}\).

Indeed, in this case the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1} + 1\). Applying Rolle's theorem we deduce that the number of zeros of \(\varphi^{(n-n_j)}\) on the interval \([x_0, x_j)\) is still \(n_0 + n_1 + \ldots + n_{j-1} + 1 \ldots\), and that the number of zeros of \(\varphi^{(n-n_0-1)}\) on the interval \([x_0, x_j)\) is still \(n_0 + n_1 + \ldots + n_{j-1}\). Then, applying Rolle's theorem again, we deduce that the number of zeros of \(\varphi^{(n-n_0)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}, \ldots\), and that the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_j)\) is

But to obtain the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_j)\), we can apply the fifth lemma to the interval \((x_0, x_l)\), since \(l \lt j\), and then the third lemma to the interval \([x_l, x_j)\). This number is at most

Since the numbers (2.4.28) and (2.4.29) are different, we must reject the hypothesis that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}\), and therefore this number is \(n_0 + n_1 + \ldots + n_{j-1} - 1\), which proves our lemma.

b) Let us consider the other two cases \(n-n_0-1 \geq n-n_l-1\). We must then take into account that, by the boundary conditions (2.1.7) at the point \(x_0\), we have

We shall prove that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is at most equal to \(n_0 - 1 + n_1 + \ldots + n_{j-1}\).

To this end let us suppose that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}\). Applying Rolle's theorem successively and taking into account conditions (2.4.30), we deduce that the number of zeros of \(\varphi^{(n-n_j)}\) on the interval \((x_0, x_j)\) is still \(n_0 + n_1 + \ldots + n_{j-1}, \ldots\) and that the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_j)\) is still

But the number of zeros of \(\varphi^{(n-n_l-1)}\) on the interval \((x_0, x_j)\) can be found by applying the fifth lemma to the interval \((x_0, x_l)\), bearing in mind that \(n-n_0-1 \geq n-n_l-1\), and by applying the third lemma to the interval \([x_l, x_j)\). This number is therefore

Since there is a disagreement between the numbers (2.4.31) and (2.4.32), we must reject the hypothesis that the number of zeros of \(\varphi^{(n-n_j-1)}\) on the interval \((x_0, x_j)\) is \(n_0 + n_1 + \ldots + n_{j-1}\). Hence it is at most \(n_0 - 1 + n_1 + \ldots + n_{j-1}\), and therefore the fifth lemma is proved in the case \(n-n_0-1 \gt n-n_j-1\) and \(n-n_0-1 \geq n-n_l-1\).

The fifth lemma is thus completely proved.

2.4.7. Let us consider all the nodes \(x_0, x_1, \ldots, x_k\) with their multiplicities \(n_0, n_1, \ldots, n_k\). Let us denote by \(n-n_i-1\) the smallest number of the sequence

and by \(x_{i_1}, x_{i_2}, \ldots, x_{i_p}\) the nodes which correspond to \(n-n_i-1\), \(x_{i_1}\) being the one closest to \(x_0\). From the preceding lemmas the following remark results:

Indeed, if \(n-n_0-1 \lt n-n_i-1\), the number of zeros of \(\varphi^{(n-n_i-1)}\) on the interval \((x_0, x_{i_r})\) is equal to the sum of the numbers of zeros of \(\varphi^{(n-n_i-1)}\) on the intervals \((x_0, x_{i_1}), [x_{i_1}, x_{i_2}), \ldots, [x_{i_{r-1}}, x_{i_r})\), which are given, for the first interval, by the fifth lemma and, for the other intervals, by the third lemma. We shall therefore have

that is,

which proves formula (2.4.34).

If \(n-n_0-1 \geq n-n_i-1\), the same reasoning shows that

and since \(n_{i_1} = n_{i_r}\), we can write

which proves formula (2.4.35).

2.4.8. Let us consider the sequence of nodes \(x_h, x_{h+1}, \ldots, x_k\) and suppose that the numbers \(n-n_{h+1}-1, n-n_{h+2}-1, \ldots, n-n_{k-1}-1\) are greater than \(n-n_h-1\).

We have:

The proof of this lemma is analogous to the proof of the fifth lemma. It is carried out by the method of complete induction, first proving the lemma analogous to the fourth lemma for the nodes \(x_{k-1}, x_k\) and taking into account the boundary conditions at the point \(x_k\). Then one assumes that the sixth lemma has been proved for the sequence of nodes \(x_h, x_{h+1}, \ldots, x_k\), where \(h \gt j\), and one proves the sixth lemma for \(h = j\).

A remark analogous to the one made in subsection 2.4.7 can be added.

2.4.9. The proof of Theorem 2.4.1 follows immediately from the preceding lemmas.

Let us denote, as above, by \(n-n_i-1\) the smallest number of the sequence \(n-n_1-1, n-n_2-1, \ldots, n-n_{k-1}-1\) and by \(x_i\) one of the points which correspond to \(n-n_i-1\).

The number of zeros of \(\varphi^{(n-n_i-1)}\) on the interval \((x_0, x_k)\) is equal to the sum of the numbers of zeros of \(\varphi^{(n-n_i-1)}\) on the intervals \((x_0, x_i)\) and \([x_i, x_k)\). It is at most equal to \(N = m + m'\), where the numbers \(m\) and \(m'\) are given by the fifth lemma and by the sixth lemma, through formulas (2.4.17) and (2.4.36).

1° If \(n-n_0-1\) and \(n-n_k-1 \lt n-n_i-1\), we have \(N = (n_1 + n_2 + \ldots + n_i) + (n_i + \ldots + n_{k-1})\), and, bearing in mind that \(n + 1 = n_0 + n_1 + \ldots + n_k\), we can write

so that formula (2.4.2) is proved.

2° If \(n-n_0-1 \lt n-n_i-1 \leq n-n_k-1\), we have \(N = (n_1 + n_2 + \ldots + n_i) + (n_{i+1} + \ldots + n_k - 1)\), that is,

and formula (2.4.3) is proved.

3° If \(n-n_k-1 \lt n-n_i-1 \leq n-n_0-1\), we have \(N = (n_0 - 1 + n_1 + \ldots + n_{i-1}) + (n_i + \ldots + n_{k-1})\), that is,

and formula (2.4.4) is proved.

4° If \(n-n_0-1 \geq n-n_i-1\) and \(n-n_k-1 \geq n-n_i-1\), we have \(N = (n_0 - 1 + n_1 + \ldots + n_{i-1}) + (n_{i+1} + \ldots + n_k - 1)\), that is,

and formula (2.4.5) is proved.

Let us denote, as in the preceding subsections, by \(n-n_i-1\) the smallest number of the sequence \(n-n_1-1, n-n_2-1, \ldots, n-n_{k-1}-1\), and let us observe that the function \(\varphi\) is continuous on the interval \((x_0, x_k)\), together with its successive derivatives up to order \(n-n_i-1\), which follows from the boundary conditions (2.1.7). Let us also write the boundary conditions at the points \(x_0, x_k\)

To prove Theorem 2.4.2, we first suppose that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\). Then, taking into account this hypothesis and the boundary conditions (2.4.38), we can determine by Rolle's theorem the number \(N'\) of zeros of the derivative \(\varphi^{(n-n_i-1)}\) on the interval \((x_0, x_k)\). But we have shown that the number of zeros of \(\varphi^{(n-n_i-1)}\) is at most equal to the number \(N\) determined by Theorem 2.4.1. We shall show that there is a contradiction between the numbers \(N\) and \(N'\), brought to light by the relation \(N' = N+1\). From this contradiction it will follow that the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\).

We shall prove the formula \(N' = N + 1\) in detail, going again through all the cases examined in subsection 2.4.9, where the proof of Theorem 2.4.1 was given.

1° Let us suppose that \(n-n_0-1\) and \(n-n_k-1 \lt n-n_i-1\). We may restrict ourselves to the case \(n-n_0-1 \lt n-n_k-1\).

Applying Rolle's theorem successively and taking into account the boundary conditions (2.4.38) and the fact that the function \(\varphi\) vanishes at a point of the interval \((x_0, x_k)\), we deduce that the derivative \(\varphi'\) vanishes at two points of the interval \((x_0, x_k)\), \(\ldots\), and that the derivative \(\varphi^{(n-n_0-1)}\) vanishes at \(n - n_0\) points of the interval \((x_0, x_k)\).

Further, the derivative \(\varphi^{(n-n_0)}\) vanishes at \(n-n_0+1\) points of the interval \((x_0, x_k)\), \(\ldots\), and the derivative \(\varphi^{(n-n_k-1)}\) vanishes at \(n - n_0 + 1\) points of the interval \((x_0, x_k)\).

Finally, the derivative \(\varphi^{(n-n_k)}\) vanishes at \(n-n_0+1\) points of the interval \((x_0, x_k)\), the derivative \(\varphi^{(n-n_k+1)}\) vanishes at \(n-n_0\) points of the interval \((x_0, x_k)\), \(\ldots\), and the derivative \(\varphi^{(n-n_i-1)}\) vanishes at \(N' = n + n_i - n_0 - n_k + 2\) points of the interval \((x_0, x_k)\).

But this is impossible because, by Theorem 2.4.1, in the case \(n-n_0-1 \lt n-n_i-1\), \(n-n_k-1 \lt n-n_i-1\), the number of zeros of \(\varphi^{(n-n_i-1)}\) on the interval \((x_0, x_k)\) is at most \(N = n + n_i - n_0 - n_k + 1\).

There is therefore a contradiction between the numbers \(N\) and \(N'\); we have \(N' = N + 1\), and therefore we must reject the hypothesis that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\).

Hence the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\).

By analogous reasoning one reaches the same conclusion when \(n-n_0-1 \geq n-n_k-1\).

2° Let us suppose that \(n-n_0-1 \lt n-n_i-1 \leq n-n_k-1\), and let us assume, as above, that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\). Then, proceeding as in the preceding case, we reach the conclusion that the derivative \(\varphi^{(n-n_i-1)}\) must vanish at \(N' = n-n_0+1\) points of the interval \((x_0, x_k)\).

But, by Theorem 2.4.1, the derivative \(\varphi^{(n-n_i-1)}\) has at most \(N = n-n_0\) zeros on the interval \((x_0, x_k)\).

There is therefore a contradiction between the numbers \(N\) and \(N'\); we have \(N' = N+1\), and therefore we must reject the hypothesis that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\).

Hence the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\).

3°. Let us suppose that \(n-n_k-1 \lt n-n_i-1 \leq n-n_0-1\) and let us assume that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\). Then, proceeding as in the preceding cases, we reach the conclusion that the derivative \(\varphi^{(n-n_i-1)}\) must vanish at \(N' = n-n_k+1\) points of the interval \((x_0, x_k)\).

But, by Theorem 2.4.1, the derivative \(\varphi^{(n-n_i-1)}\) has at most \(N = n-n_k\) zeros on the interval \((x_0, x_k)\).

There is therefore a contradiction between the numbers \(N\) and \(N'\); we have \(N' = N+1\), and therefore we must reject the hypothesis that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\).

Hence the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\).

4°. Let us suppose that \(n-n_0-1 \geq n-n_i-1\) and \(n-n_k-1 \geq n-n_i-1\), and let us assume that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\). Then, proceeding as in the preceding cases, we reach the conclusion that the derivative \(\varphi^{(n-n_i-1)}\) must vanish at \(N' = n-n_i\) points of the interval \((x_0, x_k)\).

But, by Theorem 2.4.1, the derivative \(\varphi^{(n-n_i-1)}\) has at most \(N = n-n_i-1\) zeros on the interval \((x_0, x_k)\).

There is therefore a contradiction between the numbers \(N\) and \(N'\); we have \(N' = N+1\), and therefore we must reject the hypothesis that the function \(\varphi\) vanishes at least once on the interval \((x_0, x_k)\).

Hence the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\).

We have thus proved that in all cases the function \(\varphi\) does not vanish on the interval \((x_0, x_k)\). The sign of the function \(\varphi\) can be found by observing that if in formulas (2.1.27) and (2.1.28) the function \(f(x)\) is replaced by \(x^n\), one finds \(1 = n! \int_{x_0}^{x_k} \varphi(s)\, ds\), whence it follows that the function \(\varphi\) is positive on the interval \((x_0, x_k)\) and we have the formula

Indeed, by Theorem 2.4.2 the function \(\varphi\) in the integral representation (2.1.28) is positive on the interval \((x_0, x_k)\) and we have formula (2.4.39). The mean value theorem can be applied, and we are led to formula (2.4.40).

Formula (2.4.40) shows that

where

§ 2.5. The Lagrange–Hermite interpolation formula

The representation of the divided difference of a function \(f\) on multiple nodes by a definite integral leads in a remarkable way to the Lagrange–Hermite interpolation formula, to its remainder and to the Lagrange–Hermite polynomial.

2.5.1. Let us consider a function \(f\) defined on the interval \((a, b)\) and the nodes \(x_1, x_2, \ldots, x_k\) of multiplicities \(n_1, n_2, \ldots, n_k\), such that \(a \lt x_1 \lt \ldots \lt x_k \lt b\), as well as the simple node \(x \in (a, b)\), which does not coincide with any of the nodes \(x_1, x_2, \ldots, x_k\). The node \(x\) may lie to the left of \(x_1\), to the right of \(x_k\), or in the interval \((x_1, x_k)\).

Let us first suppose that the node \(x \in (x_i, x_{i+1})\), where \(i\) is one of the numbers \(1, 2, \ldots, k-1\). Then we can consider the divided difference

and its integral representation

where the number \(n\) is given by the formula

and where we have denoted the function \(\varphi\) of formula (2.1.28) by \(\varphi_i(x, s)\), to show that it depends on the node \(x \in (x_i, x_{i+1})\).

The left-hand side of formula (2.5.2) can be replaced by

and therefore formula (2.5.2) becomes

But

so that, introducing the polynomial

we can write

Formula (2.5.5) becomes

The first term of the right-hand side is a polynomial \(H_{n-1}(x)\), which is of degree at most \(n-1\). We shall write

Concerning the polynomial \(H_{n-1}\) we have the following two theorems:

Indeed, replacing \(x\) by \(x_l\) in formula (2.5.9) and in the successive derivatives of both its sides, we can write

which shows that

and therefore Theorem 2.5.1 is proved.

Indeed, the equations which determine the coefficients of the polynomial \(P_{n-1}\) are

and the determinant of this system is different from zero.

Eliminating the coefficients \(A_{n-1}, A_{n-2}, \ldots, A_0\) between equations (2.5.11) and (2.5.14), one finds:

This equation, solved with respect to \(P_{n-1}(x)\) and compared with \(H_{n-1}(x)\), shows that \(P_{n-1} = H_{n-1}\), and therefore Theorem 2.5.2 is proved.

Formula (2.5.8) can be written in the form

It appears as an approximation formula, which is called the Lagrange–Hermite interpolation formula. The function \(f\) is approximated by the polynomial \(H_{n-1}\), which is called the Hermite interpolation polynomial.

The Hermite interpolation polynomial is given by formula (2.5.9) or by Theorem 2.5.2.

The remainder of formula (2.5.11) is:

it depends on \(x\) and on the index \(i\), which indicates that \(x \in (x_i, x_{i+1})\), where \(i\) can be \(1, 2, \ldots, k-1\).

2.5.2. If \(x \lt x_1\), we have the integral representation:

where we have denoted the function \(\varphi\) of formula (2.1.28) by \(\varphi_0(x, s)\), to show that it corresponds to the node \(x \lt x_1\).

Proceeding as in subsection 2.5.1, we arrive at the Lagrange–Hermite interpolation formula

where the remainder of this formula is:

If \(x \gt x_k\), we have the integral representation

where we have denoted the function \(\varphi\) of formula (2.1.28) by \(\varphi_k(x, s)\), to show that it corresponds to the node \(x \gt x_k\).

Proceeding as in subsection 2.5.1, we arrive at the Lagrange–Hermite interpolation formula:

where the remainder is given by the formula:

2.5.3. The function \(\varphi_i(x, s)\) is positive on the interval \((x_1, x_k)\) for \(i = 1, 2, \ldots, k-1\), just like the functions \(\varphi_0(x, s), \varphi_k(x, s)\) on the intervals \((x, x_k), (x_1, x)\). We have

and

For the remainders \(R_i(x)\) we obtain the estimates

for \(i = 0, 1, \ldots, k\), where

and

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Română

It is known that the nodes \(x_1, x_2, \ldots, x_n\) and the coefficients \(C_1, C_2, \ldots, C_n\) of the quadrature formula

can be determined so that the remainder \(R[f]\) vanishes when the function \(f(x)\) is replaced by \(1, x, \ldots, x^{2n-1}\). The nodes \(x_1, x_2, \ldots, x_n\) are the roots of the Legendre polynomial associated with the interval \([a, b]\). Formula (3.1) is called Gauss's quadrature formula.

P. Turán [47] gave a beautiful extension of Gauss's quadrature formula, proving that one can determine the nodes and the coefficients of the quadrature formula

so that the remainder \(R[f]\) vanishes when the function \(f(x)\) is replaced by \(1, x, \ldots, x^{4n-1}\).

More generally, P. Turán considered quadrature formulas of the form

and determined the nodes and the coefficients so that the remainder \(R[f]\) vanishes when the function \(f(x)\) is replaced by \(1, x, \ldots, x^{2(k+1)n-1}\).

Formulas (3.3) form a class; they are called the formulas of P. Turán. For \(k = 0\), formula (3.3) becomes Gauss's quadrature formula.

In this chapter we shall determine the remainders of the quadrature formulas of Gauss and P. Turán.

We have shown that Gauss's quadrature formula and its remainder can be derived from the divided difference

where \(x_1, x_2, \ldots, x_n\) are the roots of the Legendre polynomial associated with the interval \([a, b]\), and where we put \(g' = f\).

Using Theorem 2.4.3, one can derive an estimate of the absolute value of the remainder of Gauss's quadrature formula. Using the integral representation (2.1.28), we are led to the integral representation of the remainder \(R[f]\) of Gauss's quadrature formula.

This idea will be taken up again for P. Turán's quadrature formula (3.2) or for the general formula (3.3).

This chapter begins with the integral representation of the divided difference

by the method used in Chapter II. However, the proof concerning the sign of the function \(\varphi\) on the interval \((a, b)\) is simpler in the case of the divided difference (3.5) than in the general case. This is due to the fact that in the divided difference (3.5) the nodes \(a, b\) are simple and the nodes \(x_1, x_2, \ldots, x_n\) are multiple, all of the same multiplicity \(k\).

§ 3.1. The integral representation of the divided difference \([a, \underbrace{x_1, \ldots, x_1}_{k}, \underbrace{x_2, \ldots, x_2}_{k}, \ldots, \underbrace{x_n, \ldots, x_n}_{k}, b; f]\)

3.1.1. In view of the integral representation of the divided difference

let us consider the number \(N\) defined by \(N + 1 = 1 + kn + 1\), that is,

where \(k \gt 1\), and let us assume that \(f \in C^{kn+1}[a, b]\).

We proceed as in Chapter II and associate with the intervals \([a, x_1], [x_1, x_2], \ldots, [x_n, b]\) the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_{n+1}\), solutions of the differential equations

We have the following formulas:

Applying the generalized integration by parts formula to each integral (3.1.4), we shall have

Adding formulas (3.1.5) term by term, we obtain the formula:

where \(\mathscr{P}[f]\) depends on the values of \(f\) and of its successive derivatives \(f', f'', \ldots, f^{(N-1)}\) at the nodes \(a, x_1, \ldots, x_n, b\), and where the function \(\varphi\) coincides on the intervals \([a, x_1], [x_1, x_2], \ldots, [x_n, b]\) with the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_{n+1}\).

Concerning formula (3.1.6), we pose the following problem:

For this we must introduce the boundary conditions:

The solution of this boundary value problem is easily obtained. The differential equations (3.1.3) and the boundary conditions (3.1.8) at the points \(a, x_1, \ldots, x_n\) are satisfied by the polynomials:

where \(\lambda_0, \lambda_1, \lambda_1^{(1)}, \ldots, \lambda_1^{(k-1)}, \ldots, \lambda_n, \lambda_n^{(1)}, \ldots, \lambda_n^{(k-1)}\) are constants.

The constants \(\lambda_0, \lambda_1, \ldots, \lambda_n^{(k-1)}\) are determined by requiring that the boundary conditions (3.1.8) at the point \(b\) be satisfied. We then obtain the equations

Introducing a new unknown \(\lambda_{n+1}\) through the equation

equations (3.1.10) can be written in the form

The number of unknowns in system (3.1.12) is \(1 + kn + 1 = N + 1\) and the number of equations of the system is \(N\). The rank of the coefficient matrix of system (3.1.12) is \(N\). Denoting by \(D_0, D_1, D_1^{(1)}, \ldots, D_1^{(k-1)}, \ldots, D_n, D_n^{(1)}, \ldots, D_n^{(k-1)}, D_{n+1}\) the determinants of the matrices obtained from the matrix of system (3.1.12) by deleting the first, the second, \(\ldots\), the last column of the matrix of the system, we have

where \(H\) is a constant.

The constants \(\lambda_0, \lambda_1, \lambda_1^{(1)}, \ldots, \lambda_1^{(k-1)}, \ldots, \lambda_n, \lambda_n^{(1)}, \ldots, \lambda_n^{(k-1)}, \lambda_{n+1}\) being determined up to a factor \(H\), the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_{n+1}\) are likewise determined up to the factor \(H\). Replacing in formula (3.1.6) the polynomials \(\varphi_1, \varphi_2, \ldots, \varphi_{n+1}\) by the solution of the boundary value problem, we find for \(\mathscr{P}[f]\) the expression

But by formulas (3.1.9) we have

Formula (3.1.14) becomes

Replacing the constants \(\lambda_0, \lambda_1, \ldots, \lambda_1^{(k-1)}, \ldots, \lambda_n^{(k-1)}, \lambda_{n+1}\) by their expressions derived from formulas (3.1.13), we see that formula (3.1.16) reduces to:

It remains to take into account the last boundary condition \(\mathscr{P}[x^N] = 1\) as well. Formula (3.1.17) shows that we must have

whence it follows that \(\mathscr{P}[f]\) reduces to the divided difference

Finally, taking into account formula (3.1.6), we have:

Formula (3.1.20) can also be written in the form:

We shall now prove several theorems on the coefficients of formula (3.1.21) and on the function \(\varphi\).

For \(r = 0\) the theorem is obvious, because by formulas (3.1.13) and (3.1.18) we have

The theorem is also obvious for \(r = n\), because by the first equation of (3.1.12), and by (3.1.13) and (3.1.18), we have

and

It remains to prove the theorem for \(r = 1, 2, \ldots, n-1\).

To this end, let us introduce the interpolation polynomial \(h(x; r) = h(x)\) determined by the conditions

It is easy to show that the polynomial \(h\) is completely determined by the \(nk + 1 = N\) conditions (3.1.23). It follows that the degree of the polynomial \(h\) is at most \(N-1\).

We add that the polynomial \(h\) is not identically zero, because for any \(r = 1, 2, \ldots, n-1\) we have \(h(a) = 1\), whereas \(h(x_n) = 0\).

Replacing in formula (3.1.21) the function \(f\) by the polynomial \(h\), we have

and Theorem 3.1.2, which we have in view, amounts to proving that for the polynomial \(h\) determined by conditions (3.1.23) we have \(h(b) \neq 0\).

Let us first prove that the polynomial \(h^{(k-1)}\) is of degree \(N-k\). Applying Rolle's theorem and taking into account conditions (3.1.23), we find that the derivative \(h'\) vanishes at \(n-1 + n = 2n-1\) distinct points of the interval \([a, x_n)\), then that the derivative \(h''\) vanishes at \(2n-2+n\) distinct points of the same interval, \(\ldots\), and finally that the derivative \(h^{(k-1)}\) vanishes at \(kn - (k-1) = N-k\) distinct points of the same interval.

We have thus exhibited, for the polynomial \(h^{(k-1)}\) of degree at most \(N-k\), \(N-k\) distinct roots in the interval \([a, x_n]\). This will help us to show that the degree of the polynomial \(h^{(k-1)}\) is \(N-k\).

Indeed, let us assume the contrary, namely that the degree of the polynomial \(h^{(k-1)}\) is less than \(N-k\). Then, since this polynomial has \(N-k\) distinct roots, it is identically zero, and \(h^{(k-2)}\) is a constant, which is zero because \(h^{(k-2)}(x_1) = 0\). One shows in the same way that the derivative \(h^{(k-3)}\) is zero, \(\ldots\), and finally that \(h'\) is zero. It would follow that the polynomial \(h\) is a constant, which is impossible because \(h(a) = 1\) and \(h(x_n) = 0\).

Hence the hypothesis that the degree of the derivative \(h^{(k-1)}\) is less than \(N-k\) must be rejected, and therefore the degree of \(h^{(k-1)}\) is \(N-k\).

We can now complete the proof and show that \(h(b) \neq 0\). Let us assume the contrary, namely that \(h(b) = 0\). Then, repeating the preceding arguments, we arrive at the conclusion that the successive derivatives \(h', h'', \ldots, h^{(k-1)}\) must have one more root in the interval \((x_n, b)\). But this is impossible, because it has been shown that the polynomial \(h^{(k-1)}\) of degree \(N-k\) has \(N-k\) distinct roots in the interval \([a, x_n]\). Hence the hypothesis that \(h(b) = 0\) must be rejected, and therefore \(h(b) \neq 0\).

Theorem 3.1.2 is thus proved.

Indeed, by formulas (3.1.9) the coefficients of \(s^{N-1}\) in these polynomials are

and by Theorem 3.1.2 they are all different from zero.

Indeed, by formulas (3.1.9) and the boundary conditions (3.1.8) at the point \(b\), we have \(\varphi_1(s) = \lambda_0 \frac{(s-a)^{N-1}}{(N-1)!}\), \(\varphi_{n+1}(s) = -\lambda_{n+1} \frac{(s-b)^{N-1}}{(N-1)!}\), and therefore the derivative \(\varphi^{(N-k-1)}\) does not vanish on the intervals \((a, x_1], [x_n, b)\).

On the other hand, the derivatives \(\varphi_2^{(N-k-1)}, \varphi_3^{(N-k-1)}, \ldots, \varphi_n^{(N-k-1)}\) are polynomials of degree \(k\), and the coefficients of \(\frac{x^k}{k!}\) in them are \(\lambda_0 + \lambda_1, \lambda_0 + \lambda_1 + \lambda_2, \ldots, \lambda_0 + \lambda_1 + \ldots + \lambda_{n-1}\), all different from zero by Theorem 3.1.3. It follows that the derivative \(\varphi^{(N-k-1)}\) has at most \(k\) zeros on each of the intervals \((x_1, x_2], \ldots, (x_{n-1}, x_n)\), and therefore the derivative \(\varphi^{(N-k-1)}\) has at most \(k(n-1)\) zeros in the interval \((a, b)\).

Theorem 3.1.4 is therefore proved.

To prove this theorem, let us first show that the function \(\varphi\) does not vanish on the interval \((a, b)\). Let us assume the contrary, namely that the function \(\varphi\) vanishes at least at one point of the interval \((a, b)\), and let us observe that, by the boundary conditions (3.1.8), the function \(\varphi\) and its successive derivatives \(\varphi', \varphi'', \ldots, \varphi^{(N-k-1)}\) are continuous on the interval \([a, b]\). Then, applying Rolle's theorem successively, we deduce that the derivative \(\varphi'\) must have two zeros in the interval \((a, b)\), that the derivative \(\varphi''\) must have three zeros in the interval \((a, b)\), \(\ldots\), and that the derivative \(\varphi^{(N-k-1)}\) must have \(N-k = k(n-1) + 1\) zeros in the interval \((a, b)\). But this is impossible, because by Theorem 3.1.4 the derivative \(\varphi^{(N-k-1)}\) has at most \(k(n-1)\) zeros in the interval \((a, b)\). Hence the hypothesis that the function \(\varphi\) vanishes at least at one point of the interval \((a, b)\) must be rejected, and therefore the function \(\varphi\) does not vanish on the interval \((a, b)\).

If in formula (3.1.20) we replace the function \(f\) by \(x^N\), we deduce that \(1 = N! \int_a^b \varphi\, ds\), whence it follows that the function \(\varphi\) is positive on the interval \((a, b)\) and that formula (3.1.24′) holds.

§ 3.2. The quadrature formula of Gauss

3.2.1. Let us consider Gauss's quadrature formula of the form

where \(p\) is a weight, a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\).

We shall then have the identity

for \(f(x) = x^k\), where \(k = 0, 1, \ldots, 2n-1\).

We must require that formula (3.2.2) become an identity when the function \(f(x)\) is replaced by

and

We may also say that formula (3.2.2) holds when the function \(f\) is replaced by an arbitrary polynomial \(Q\) of degree at most \(2n-1\).

Let us denote by

the polynomial whose roots are the nodes of formula (3.2.2). To determine the nodes \(x_1, x_2, \ldots, x_n\) means to determine the coefficients of the polynomial \(\omega_n(x)\).

The polynomials

where \(h = 0, 1, \ldots, n-1\), are of degree \(n + h \leq 2n-1\).

Replacing in formula (3.2.2) the function \(f\) by the polynomials \(Q\), we obtain the equations

because the polynomial \(\omega_n(x)\) vanishes for \(x_1, x_2, \ldots, x_n\).

As is well known, equations (3.2.7) determine the polynomial \(\omega_n(x)\), and from these equations one deduces the fundamental orthogonality property:

which means that the polynomials \(\omega_m\) form a sequence of polynomials orthogonal with respect to the function \(p(x)\) on the interval \([a, b]\), where \(m = 0, 1, \ldots\) and \(\omega_0(x) = 1\).

It is known that the polynomial \(\omega_n(x)\), whose coefficients are determined by equations (3.2.7), has all its roots real, distinct and lying between \(a\) and \(b\).

Thus the nodes of formula (3.2.2) are the roots of the polynomial \(\omega_n\). In particular, when \(p(x) = 1\), the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.2.2) are the roots of the Legendre polynomial of degree \(n\) attached to the interval \([a, b]\).

Once the nodes of formula (3.2.2) have been determined, it is easy to determine the coefficients \(C_1, C_2, \ldots, C_n\). It suffices to require that formula (3.2.2) become an identity when the function \(f\) is replaced by the functions (3.2.3). The system of equations

determines the coefficients \(C_1, C_2, \ldots, C_n\), its determinant being the Vandermonde determinant of the distinct nodes \(x_1, x_2, \ldots, x_n\).

Thus Gauss's problem is completely solved [9].

Another argument leading to these results can be found in paper [8].

Gauss's problem leads us to the identities:

obtained by adjoining to equations (3.2.9) the identities that result from replacing in formula (3.2.2) the function \(f\) by the functions (3.2.4).

It now remains to determine the remainder \(R[f]\) of Gauss's quadrature formula, when the nodes and the coefficients have been determined by Gauss's problem and when it is assumed that \(f \in C^{2n}[a, b]\). This problem was treated by us in general, with the method of the function \(\varphi\), for a weight \(p(x)\) integrable on the interval \([a, b]\) and positive on the interval \((a, b)\), in our paper [8].

However, when \(p(x) = 1\), the determination of the remainder \(R[f]\) can be connected with the integral representation of a divided difference, as we shall see below.

3.2.2. Let us consider the divided difference of the function \(g\) on the simple nodes \(a, b\) and the distinct double nodes \(x_1, x_2, \ldots, x_n\):

As is well known, we have

where \(V\) is the generalized Vandermonde determinant \(V(a, x_1, x_1, \ldots, x_n, x_n, b)\).

Expanding the determinant on the right-hand side of formula (3.2.12) along the elements of the last column, we have

where

We have the following theorem:

It suffices to carry out the proof for one of these coefficients, for example for \(A_1\). To this end, we write \(A_1\) in the form

where \(L_1\) is a coefficient that differs from \(V^{-1}\) by a factor of 1 or \(-1\).

Adding the elements of the last column to the elements of the first column, and placing the factors \(1, 2, \ldots, 2n\) beside \(L_1\), we can write \(A_1\) in the form:

The elements of the first \(n+1\) columns satisfy one and the same linear relation, as follows from identities (3.2.10). It follows that \(A_1 = 0\), and therefore Theorem 3.2.1 is proved.

3.2.3. This being so, let us replace in formula (3.2.13) the nodes \(x_1, x_2, \ldots, x_n\) by the roots of the Legendre polynomial associated with the interval \([a, b]\). By Theorem 3.2.1 we shall have \(A_1 = 0, \ldots, A_n = 0\), and we can write

Replacing in this identity the function \(g\) by 1, we deduce that

On the other hand, we have

It follows that formula (3.2.16) can be written in the form:

Let us replace in this identity the function \(g\) by:

and divide both sides by \(A_{n+1}\). We shall have the formula:

which is precisely Gauss's quadrature formula, with the nodes \(x_1, x_2, \ldots, x_n\), the roots of the Legendre polynomial associated with the interval \([a, b]\), and with the coefficients

and the remainder

If we use the integral representation of the divided difference on the right-hand side of formula (3.2.23), we obtain the integral representation of the remainder \(R[f]\), when \(f \in C^{2n}[a, b]\):

where on the right-hand side there appears the function \(\varphi\) of the integral representation of the divided difference (3.2.11) corresponding to the nodes \(x_1, x_2, \ldots, x_n\), the roots of the Legendre polynomial associated with the interval \([a, b]\).

We thus have:

3.2.4. Introducing the polynomial

where \(x_1, x_2, \ldots, x_n\) are the roots of the Legendre polynomial associated with the interval \([a, b]\), and replacing in formula (3.2.21) the function \(f\) by \(\omega_n^2\), we obtain

But \(\int_a^b \varphi\, ds = \frac{1}{(2n+1)!}\), and therefore we deduce from the preceding formula

which is an interesting interpretation of the coefficient \(\frac{1}{A_{n+1}}\) on the right-hand side of formula (3.2.24).

§ 3.3. The quadrature formula of P. Turán

3.3.1. Let us consider the quadrature formula of the form

where \(p\) is a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\), and where the right-hand side contains the derivatives \(f^{(l)}(x)\) at the nodes \(x_1, x_2, \ldots, x_n\) for \(l = 0, 1, \ldots, 2(s-1)\). This formula has been the subject of important investigations by P. Turán [47] and is called P. Turán's quadrature formula.

We shall then have the identity

for \(f(x) = x^k\), \(k = 0, 1, \ldots, 2ns-1\).

P. Turán showed that this problem is connected with a minimum problem, namely: to determine the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) so that the integral

where \(s\) is a natural number, is minimal.

The polynomial that realizes the minimum of integral (3.3.3) is called the minimizing polynomial of the integral. Below we shall give a simple proof of Theorem 3.3.1, due to Margareta Frenkel [4]. The minimizing polynomial of integral (3.3.3):

enjoys important properties, analogous to the properties of the Legendre polynomials.

We shall first prove:

For \(s = 1\), equations (3.3.5) express the property of the Legendre polynomials of being orthogonal with respect to the interval \((a, b)\) and the weight \(p\).

Next we shall prove:

The minimizing polynomial \(\omega_n\) is very important for P. Turán's problem, because we have:

Finally, we shall show that the remainder \(R[f]\) of the quadrature formula (3.3.1) of P. Turán, when \(f \in C^{2ns}[a, b]\) and \(p(x) = 1\), can be connected with the integral representation of a divided difference, as we did for the remainder of Gauss's formula [10].

3.3.2. Proof of D. Jackson's Theorem 3.3.1, given by Margareta Frenkel. Let us consider the integral

where \(p\) is a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\), \(s\) is a natural number, and \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are real coefficients. We shall determine the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) uniquely so that the integral \(\delta_{2s}(\lambda_1, \lambda_2, \ldots, \lambda_n)\) is minimal.

To this end, we shall consider the function

and observe that it has the following properties:

1° We have

2° \(w(\lambda_0, \lambda_1, \ldots, \lambda_n)\) is a homogeneous polynomial of degree \(2s\), which means that it can be written in the form:

where \(i_0 + i_1 + \ldots + i_n = 2s\).

3° We have

if and only if \(\lambda_0^2 + \lambda_1^2 + \ldots + \lambda_n^2 = 0\).

These properties are obvious.

4° One can determine two positive numbers \(\alpha\) and \(\beta\), \(\alpha \lt \beta\), such that

where

Indeed, let us consider the set of points of \(\mathbb{R}^{n+1}\) lying on the sphere \(S_1\) of radius 1, that is, the set of systems of numbers \((\mu_0, \mu_1, \ldots, \mu_n)\) for which \(|\mu| = 1\). Since this set is bounded and closed, and the function \(w(\lambda_0, \lambda_1, \ldots, \lambda_n)\) considered on this sphere is continuous, we can determine the numbers

The number \(\alpha\) is positive, because the function \(w(\lambda_0, \lambda_1, \ldots, \lambda_n)\) is positive on the sphere and its \(\inf\) is attained on the sphere; hence it is positive.

We shall have

Given an arbitrary system of real numbers \((\lambda_0, \lambda_1, \ldots, \lambda_n)\), one can determine a system of numbers \((\mu_0, \mu_1, \ldots, \mu_n)\), where \(|\mu| = 1\), and a positive number \(l\) such that

Since the function \(w(\lambda_0, \lambda_1, \ldots, \lambda_n)\) is a homogeneous polynomial of degree \(2s\), we have \(w(\lambda_0, \lambda_1, \ldots, \lambda_n) = l^{2s} w(\mu_0, \mu_1, \ldots, \mu_n)\), and by inequalities (3.3.15) we have

Formulas (3.3.16) show that \(l = |\lambda| = \sqrt{\lambda_0^2 + \lambda_1^2 + \ldots + \lambda_n^2}\), and therefore inequalities (3.3.17) can be written in the form

or else in the form

This being so, let us set \(\lambda_0 = 1\) in integral (3.3.7). Then this integral becomes a function \(w(1, \lambda_1, \ldots, \lambda_n)\) of \(\lambda_1, \lambda_2, \ldots, \lambda_n\):

which is fundamental for this proof and will be studied below.

Inequalities (3.3.19) become

With the function \(w(1, \lambda_1, \ldots, \lambda_n)\) we shall associate the number \(\nu\) defined by the equation

We shall have

The function \(w(1, \lambda_1, \ldots, \lambda_n)\) has several properties:

Indeed, by inequalities (3.3.21) we have \(w(1, \lambda_1, \ldots, \lambda_n) \geq \alpha(1 + \lambda_1^2 + \ldots + \lambda_n^2)^s\), and for \(\lambda_1^2 + \ldots + \lambda_n^2 \gt \nu^2\) we shall have \(w(1, \lambda_1, \ldots, \lambda_n) \gt \alpha(1 + \nu^2)^s = \alpha \dfrac{\beta}{\alpha} = \beta\); therefore the first property is proved.

To prove this property, let us first observe that the point \((0, 0, \ldots, 0)\) belongs to the set of points \((\lambda_1, \ldots, \lambda_n)\) for which \(\lambda_1^2 + \ldots + \lambda_n^2 \leq \nu^2\), and inequalities (3.3.21) show that \(w(1, 0, \ldots, 0) \leq \beta\). By the definition of \(w(1, \lambda_1, \ldots, \lambda_n)\) we have

for every point \((\lambda_1, \ldots, \lambda_n)\) for which \(\lambda_1^2 + \ldots + \lambda_n^2 \lt \nu^2\). In particular, we shall have

and thus the third property is proved.

To see this, we observe that if we consider the function \(w(1, \lambda_1, \ldots, \lambda_n)\) for \(\lambda_1^2 + \ldots + \lambda_n^2 \geq \nu^2\), we have by the first property \(w(1, \lambda_1, \ldots, \lambda_n) \gt \beta\).

But if we consider the function \(w(1, \lambda_1, \ldots, \lambda_n)\) for \(\lambda_1^2 + \ldots + \lambda_n^2 \leq \nu^2\), it has an inf attained at \((\lambda_1^0, \ldots, \lambda_n^0)\), and by the third property we have

Hence \(I\) is the inf of the function \(w(1, \lambda_1, \ldots, \lambda_n)\) for \((\lambda_1, \ldots, \lambda_n) \in \mathbb{R}^n\), and it is attained at the point \((\lambda_1^0, \ldots, \lambda_n^0) \in \mathbb{R}^n\), because, by the first property, \(I\) cannot be attained at points \((\lambda_1, \ldots, \lambda_n)\) for which \(\lambda_1^2 + \ldots + \lambda_n^2 \gt \nu^2\).

Indeed, the function \(w(1, \lambda_1, \ldots, \lambda_n)\) is defined at every point \((\lambda_1, \ldots, \lambda_n) \in \mathbb{R}^n\), and the point \((\lambda_1^0, \ldots, \lambda_n^0)\) at which the function attains its inf is an interior point. It follows that \((\lambda_1^0, \ldots, \lambda_n^0)\) is a minimum point, and therefore we have formulas (3.3.27).

This property is connected with the inequality

valid for \(s = 1, 2, \ldots\) and \(u \neq v\), which is a particular case of the inequality

where \(A\) and \(B\) are two positive real numbers, \(A + B = 1\), and \(u \neq v\).

Inequality (3.3.28) is easily proved.

This being so, let us assume that the point \((\lambda_1^0, \ldots, \lambda_n^0)\) at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) attains its inf is not unique, which means that there exist at least two different points \((\lambda_1^0, \ldots, \lambda_n^0)\) and \((\lambda_1', \ldots, \lambda_n')\) at which the inf is attained. Then we have

The polynomials

are not identical. They may coincide at a certain number of points \(c_1, c_2, \ldots, c_q\) of the interval \([a, b]\), where \(q \leq n\).

On the intervals \((a, c_1), (c_1, c_2), \ldots, (c_q, b)\) we shall have the inequalities

as inequality (3.3.28) shows.

Taking into account formulas (3.3.31), we can write

valid on each interval \((a, c_1), (c_1, c_2), \ldots, (c_q, b)\).

Multiplying both sides of this inequality by \(p(x)\), where \(x \in (c_i, c_{i+1})\), and integrating from \(c_i\) to \(c_{i+1}\), we have

We may consider inequalities (3.3.32) for \(i = 0, 1, \ldots, q\), where \(c_0 = a\), \(c_{q+1} = b\).

Adding all these inequalities term by term, we find

that is,

But we have assumed that

and it follows from inequality (3.3.33) that

which contradicts the definition of the inf of \(w\).

From this contradiction follows the uniqueness of the point \((\lambda_1^0, \ldots, \lambda_n^0)\) at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) attains its inf.

Let us assume that there exists a point \((\lambda_1', \ldots, \lambda_n')\) different from \((\lambda_1^0, \ldots, \lambda_n^0)\) at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has a relative minimum. This means that there exists a number \(\gamma \gt 0\) such that for every point \((\lambda_1, \ldots, \lambda_n)\) in a \(\gamma\)-neighborhood of the point \((\lambda_1', \ldots, \lambda_n')\) we have

Let us consider the polynomials \(\varphi_{\lambda^0}(x) = x^n + \lambda_1^0 x^{n-1} + \ldots + \lambda_n^0\), \(\varphi_{\lambda^1}(x) = x^n + \lambda_1' x^{n-1} + \ldots + \lambda_n'\) and \(\varphi(x) = A\varphi_{\lambda^0}(x) + B\varphi_{\lambda^1}(x)\), where \(A\) and \(B\) are two positive numbers, \(A + B = 1\). The numbers \(A\), \(B\) will be subjected to further conditions, which we shall specify later.

We can write

Since the points \((\lambda_1^0, \ldots, \lambda_n^0)\), \((\lambda_1', \ldots, \lambda_n')\) are distinct, the polynomials \(\varphi_{\lambda^0}\), \(\varphi_{\lambda^1}\) are not identical. They may coincide at a certain number of points \(d_1, \ldots, d_q\). Let us put \(d_0 = a\), \(d_{q+1} = b\) and apply inequality (3.3.29) on each interval \((a, d_1), (d_1, d_2), \ldots, (d_q, b)\). We shall have

Multiplying both sides by \(p(x)\), then integrating from \(d_i\) to \(d_{i+1}\) and adding these inequalities term by term, we shall have

Let us now consider a positive number \(\varepsilon \lt \gamma\) and choose the number \(A\) such that

Then the point \((A\lambda_1^0 + B\lambda_1', \ldots, A\lambda_n^0 + B\lambda_n')\) belongs to the neighborhood of \((\lambda_1', \ldots, \lambda_n')\), where \(\varepsilon \lt \gamma\), because

Let us return to inequality (3.3.35), which we write in the form

or else in the form

But since \((\lambda_1^0, \ldots, \lambda_n^0)\) is the point at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) attains its inf, the large bracket on the left-hand side is positive. On the left-hand side of inequality (3.3.37) both terms are positive, and it follows that for a number \(A\) satisfying the inequalities \(A \lt 1\) and \(A|\lambda_i' - \lambda_i^0| \lt \varepsilon\), \(i = 1, 2, \ldots, n\), we have the inequality \(w(1, A\lambda_1^0 + B\lambda_1', \ldots, A\lambda_n^0 + B\lambda_n') \lt w(1, \lambda_1', \ldots, \lambda_n')\), which contradicts the hypothesis that the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has a minimum at \((\lambda_1', \ldots, \lambda_n')\).

We must reject this hypothesis, and therefore the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has no relative minimum other than at the point \((\lambda_1^0, \ldots, \lambda_n^0)\), where it attains its inf.

Let us assume that there exists a point \((\lambda_1'', \ldots, \lambda_n'') \in \mathbb{R}^n\) at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has a maximum. This means that there would exist a positive number \(\gamma\) such that for every point \((\lambda_1, \ldots, \lambda_n)\) in a \(\gamma\)-neighborhood of \((\lambda_1'', \ldots, \lambda_n'')\) we have

This inequality holds for all points \((\lambda_1, \ldots, \lambda_n)\) satisfying the conditions

Let us proceed as in the proof of the seventh property and consider the polynomials \(\varphi_{\lambda^0}(x) = x^n + \lambda_1^0 x^{n-1} + \ldots + \lambda_n^0\); \(\varphi_\lambda(x) = x^n + \lambda_1 x^{n-1} + \ldots + \lambda_n\), where \((\lambda_1^0, \ldots, \lambda_n^0)\) is the point at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) attains its inf, and \((\lambda_1, \ldots, \lambda_n)\) is a point that will be specified later.

Let us also consider the polynomial

where \(A\), \(B\) are two positive numbers, \(A + B = 1\), and where \(A\) will be subjected to further conditions, which we shall specify later. We have

Let us assume that the point \((\lambda_1, \ldots, \lambda_n)\) is given by the equations

that is,

so that

Since the points \((\lambda_1^0, \ldots, \lambda_n^0)\), \((\lambda_1, \ldots, \lambda_n)\) are distinct, the polynomials \(\varphi_{\lambda^0}\), \(\varphi_\lambda\) are not identical. Proceeding as in the proof of the seventh property and using inequality (3.3.29), we deduce the inequality

This being so, let \(\varepsilon\) be a positive number, \(\varepsilon \lt \gamma\), and let us show that we can impose conditions on \(A\) such that the point \((\lambda_1, \ldots, \lambda_n)\) lies in a \(\gamma\)-neighborhood of the point \((\lambda_1'', \ldots, \lambda_n'')\). Formulas (3.3.42) show that

In order that the point \((\lambda_1, \ldots, \lambda_n)\) lie in an \(\varepsilon\)-neighborhood of the point \((\lambda_1'', \ldots, \lambda_n'')\), we shall take \(A\) such that

that is,

and we observe that \(A \lt 1\).

Returning to inequality (3.3.45), we can write \(w(1, \lambda_1'', \ldots, \lambda_n'') \lt Aw(1, \lambda_1^0, \ldots, \lambda_n^0) + (1 - A)\, w(1, \lambda_1, \ldots, \lambda_n)\), or else

But the large bracket is positive, because \((\lambda_1^0, \ldots, \lambda_n^0)\) is the point at which the function \(w(1, \lambda_1, \ldots, \lambda_n)\) attains its inf.

It then follows from inequality (3.3.47) that at the point \((\lambda_1, \ldots, \lambda_n)\) we have \(w(1, \lambda_1'', \ldots, \lambda_n'') \lt w(1, \lambda_1, \ldots, \lambda_n)\), which is contrary to the hypothesis that the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has a maximum at the point \((\lambda_1'', \ldots, \lambda_n'')\), a hypothesis expressed by inequalities (3.3.38).

We must therefore reject this hypothesis, and consequently the function \(w(1, \lambda_1, \ldots, \lambda_n)\) has no maximum.

The eighth property is thus proved.

By means of the properties of the function

it has been shown that it has a unique minimum, equal to its inf and attained at the point \((\lambda_1^0, \ldots, \lambda_n^0)\).

The coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the minimizing polynomial are determined uniquely by equations (3.3.27), that is, by the equations

which can be written in the form

where \(\omega_n(x) = x^n + \lambda_1 x^{n-1} + \ldots + \lambda_n\) is the minimizing polynomial.

Thus Theorems 3.3.1 and 3.3.2 are proved.

3.3.3. Proof of Theorem 3.3.3. Once the polynomial

has been determined by equations (3.3.5)

where \(h = 0, 1, \ldots, n - 1\), which determine its coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\), we shall prove that it has all its roots real, distinct and lying between \(a\) and \(b\).

Let us first prove that the polynomial \(\omega_n(x)\) has all its roots real and lying between \(a\) and \(b\). To this end, we observe that from equations (3.3.5) we can deduce the identity

valid for every polynomial of degree at most \(n-1\), and we take into account that, by our hypotheses, the function \(p(x)\) is positive on the interval \((a, b)\).

If we put \(Q(x) = 1\) in identity (3.3.51), we must have

which proves that the polynomial \(\omega_n(x)\) has at least one root \(x_1\) lying between \(a\) and \(b\), since otherwise identity (3.3.52) would be impossible.

Suppose we have proved that the polynomial \(\omega_n(x)\) has \(x_1, x_2, \ldots, x_k\) roots lying between \(a\) and \(b\), \(k \lt n\), which is true for \(k = 1\). Then let us write

where \(\pi_{n-k}(x)\) is a polynomial of degree \(n-k\). Let us prove that the polynomial \(\pi_{n-k}(x)\) must have at least one real root lying between \(a\) and \(b\).

Indeed, let us replace in identity (3.3.51) the polynomial \(\omega_n(x)\) by formula (3.3.53); then, choosing \(Q(x) = (x - x_1) \ldots (x - x_k)\), we can write

which proves that the polynomial \(\pi_{n-k}(x)\) has at least one real root \(x_{k+1}\) lying between \(a\) and \(b\), since otherwise identity (3.3.54) would be impossible.

The preceding argument proves that all the roots of the polynomial \(\omega_n(x)\) are real and lie between \(a\) and \(b\).

Let us now prove that the roots of the polynomial \(\omega_n(x)\) are distinct.

To this end, let us assume that the polynomial \(\omega_n(x)\) has two equal roots \(x'\), \(x''\), which allows us to write

where \(\pi_2(x)\) is a polynomial of degree \(n - 2\).

If we choose \(Q(x) = \pi_2(x)\) in identity (3.3.51), we can write this identity in the form

which is impossible.

Hence all the roots of the polynomial \(\omega_n(x)\) are distinct, and Theorem 3.3.3 is proved.

3.3.4. We shall give the proof of P. Turán's Theorem 3.3.4 on the nodes and the coefficients of the formula

This formula must be valid for \(f(x) = x^k\), where \(k = 0, 1, \ldots, 2ns - 1\).

To determine the nodes and the coefficients of formula (3.3.57), we must require that this formula become an identity when we replace the function \(f\) by

and

We may also say that formula (3.3.57) holds when the function \(f\) is replaced by an arbitrary polynomial \(Q\) of degree at most \(2sn - 1\).

Let us denote by

the polynomial whose roots are the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.3.57). To determine the nodes \(x_1, x_2, \ldots, x_n\) means to determine the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the polynomial \(\omega_n(x)\).

The polynomials

where \(h = 0, 1, \ldots, n - 1\), are of degree \((2s - 1)n + h \leq 2sn - 1\). The polynomial \(Q(x)\) can also be written in the form

and one sees that \(Q(x), Q'(x), \ldots, Q^{(2(s-1))}(x)\) vanish for \(x = x_1, \ldots, x = x_n\). It follows that if in formula (3.3.57) the function \(f\) is replaced by the polynomial (3.3.61) or (3.3.62), we obtain the equations

for determining the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the polynomial \(\omega_n(x)\). This system is identical with system (3.3.5) and was studied in the proofs of Theorems 3.3.1 and 3.3.2. It was shown that this system has a unique solution, which gives the minimizing polynomial of the integral

Moreover, it was proved in Theorem 3.3.3 that the polynomial \(\omega_n(x)\) has real, distinct roots lying between \(a\) and \(b\).

Hence the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.3.57) are the roots of the minimizing polynomial \(\omega_n(x)\) of integral (3.3.3).

Thus the first part of Theorem 3.3.4 is proved.

It remains to determine the coefficients of formula (3.3.57). For this it suffices to write that formula (3.3.57) is satisfied when the function \(f\) is replaced by the functions (3.3.58).

We then find the system of linear equations

This is a system of \((2s-1)n\) equations in \((2s-1)n\) unknowns \(C_1^{(0)}, C_1^{(1)}, \ldots, C_1^{(2s-2)}, \ldots, C_n^{(0)}, C_n^{(1)}, \ldots, C_n^{(2s-2)}\), whose determinant is the generalized Vandermonde determinant \(V(\underbrace{x_1, \ldots, x_1}_{2s-1}, \ldots, \underbrace{x_n, \ldots, x_n}_{2s-1})\), which is different from zero because the nodes \(x_1, x_2, \ldots, x_n\) are distinct.

The second part of Theorem 3.3.4 is therefore proved. The coefficients of formula (3.3.57) are given by the system of equations (3.3.64).

3.3.5. The remainder of P. Turán's quadrature formula when \(p(x) = 1\). Let us suppose that in P. Turán's quadrature formula (3.3.1) we have \(p(x) = 1\). Then the remainder of this formula can be determined by a method analogous to the one used to determine the remainder of the Gauss formula (§ 3.2).

To this end let us consider the divided difference of the function \(g\) on the simple nodes \(a, b\) and on the distinct nodes \(x_1, x_2, \ldots, x_n\), each of multiplicity \(2s\):

We have

where \(V\) is the generalized Vandermonde determinant,

Let us expand the determinant on the right-hand side of formula (3.3.66) along the elements of the last column. We shall have

where \(A_1^{(0)}, A_2^{(0)}, \ldots, A_n^{(0)}\) are the coefficients in which we are particularly interested in this work. We have

Let

be the minimizing polynomial of the integral

which was studied in subsection 3.3.2 and whose roots are real, distinct and lie between \(a\) and \(b\).

We have:

It suffices to give the proof for one of these coefficients, for example for \(A_1^{(0)}\).

To this end we write the coefficient \(A_1^{(0)}\) in the form

where \(L_1^{(0)}\) is a coefficient which differs from \(V^{-1}\) by a factor of 1 or \(-1\).

Adding the elements of the last column to the elements of the first column and placing next to \(L_1^{(0)}\) the factor \(1, 2, \ldots, 2ns\), we can write \(A_1^{(0)}\) in the form

But if we write that P. Turán's formula (3.3.57) is verified when the function \(f\) is replaced by the functions (3.3.58) and (3.3.59), we have the identities

Then among the elements of the first \((2s-1)n + 1\) columns of the determinant in formula (3.3.72) there is one and the same linear combination, given by identity (3.3.73), and therefore \(A_1^{(0)} = 0\).

Thus Theorem 3.3.5 is proved.

Taking Theorem 3.3.5 into account, formula (3.3.68) becomes, when \(x_1, x_2, \ldots, x_n\) are the roots of the minimizing polynomial \(\pi_n\) of the integral (3.3.71):

If in this formula the function \(g\) is replaced by 1, we see that

On the other hand, we have

and therefore formula (3.3.74) can be written in the form

Replacing in this identity the function \(g\) by

and dividing both sides by \(A_{n+1}\), we obtain the formula

which is precisely P. Turán's quadrature formula, in which

and the remainder \(R[f]\) is

If we use the integral representation of the divided difference on the right-hand side of formula (3.3.81), when \(f \in C^{2ns}[a, b]\), we obtain the integral representation of the remainder \(R[f]\):

where the right-hand side contains the function \(\varphi\) of the integral representation of the divided difference (3.3.65) corresponding to the nodes \(x_1, x_2, \ldots, x_n\), which are the roots of the minimizing polynomial of the integral (3.3.71).

We have thus obtained:

3.3.6. Replacing in P. Turán's quadrature formula the function \(f(x)\) by \(\pi_n^{2s}(x) = (x - x_1)^{2s}(x - x_2)^{2s} \ldots (x - x_n)^{2s}\), and taking into account expression (3.3.82) of the remainder \(R[f]\), we have

But

and therefore we have:

which relates the coefficient \(\frac{1}{A_{n+1}}\) of formula (3.3.82) to the minimizing polynomial \(\pi_n\) of the integral (3.3.71).

3.3.7. The problem of determining the remainder \(R[f]\) also arises for P. Turán's quadrature formulas (3.3.1) with weight \(p\). This problem was solved by Paraschiva Pavel [28], who used our method of investigation, the method of the function \(\varphi\).

§ 3.4. Extension of the Gauss quadrature formula

3.4.1. In an earlier paper [6] we studied the quadrature formula of the form

which we regard as an extension of the Gauss quadrature formula, or else as a Gauss-type formula. In this formula \(\pi\) is a weight, a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\). Concerning this formula we shall first treat the following:

We shall then have the identity

for \(f(x) = x^k\), where \(k = 0, 1, \ldots, 2n + p + q - 1\).

We have to write that formula (3.4.2) becomes an identity when the function \(f\) is replaced by

We can also write that formula (3.4.2) holds when the function \(f\) is replaced by an arbitrary polynomial \(Q\) of degree at most \(2n + p + q - 1\).

Let us denote by

the polynomial whose roots are the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.4.2), and let us observe that determining \(x_1, x_2, \ldots, x_n\) is the same as determining the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the polynomial \(\omega_n(x)\).

The polynomials

where \(h = 0, 1, \ldots, n-1\), have degree \(n + p + q + h \leq 2n + p + q - 1\).

Replacing in formula (3.4.2) the function \(f\) by the polynomial \(Q\), we deduce the equations

because the polynomial \(\omega_n\) vanishes for \(x_1, x_2, \ldots, x_n\).

As is well known, the system formed by equations (3.4.8), where \(h = 0, 1, \ldots, n-1\), determines the polynomial \(\omega_n\), and equations (3.4.8) show that we have the fundamental orthogonality property

for \(n \neq m\), which means that the polynomials \(\omega_n\) form a sequence of orthogonal polynomials with respect to the weight \(\pi(x) (x-a)^p (b-x)^q\) and the interval \([a, b]\), where \(m = 0, 1, \ldots\) and \(\omega_0(x) = 1\).

It is known that the polynomial \(\omega_n\) is determined by the coefficients (3.4.6), which are given by equations (3.4.8). All its roots are real, distinct and lie between \(a\) and \(b\).

Thus the first part of our problem is solved; the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.4.2) are the roots of the polynomial \(\omega_n\).

Once the nodes \(x_1, x_2, \ldots, x_n\) have been determined, the coefficients \(C_1, C_2, \ldots, C_n\) follow at once. For this, the function \(f\) in identity (3.4.2) is replaced by the polynomials

where \(h = 0, 1, \ldots, n-1\).

We obtain the system of equations

which determines \(C_1, C_2, \ldots, C_n\).

Indeed, the determinant of system (3.4.11) is

It is different from zero, because the roots \(x_1, x_2, \ldots, x_n\) of the polynomial \(\omega_n\) are real, distinct and lie between \(a\) and \(b\).

Thus the coefficients \(C_1, C_2, \ldots, C_n\) of formula (3.4.2) are determined by the system of equations (3.4.11).

We may add that the coefficients \(C_1, C_2, \ldots, C_n\) are positive.

Indeed, replacing in identity (3.4.2) the function \(f\) by the polynomial \(Q(x) = (x - a)^p (b - x)^q \left(\frac{\omega_n(x)}{x - x_i}\right)^2\) of degree \(2n + p + q - 2\), we have the equation

which determines \(C_i\) and shows that \(C_i \gt 0\), \(i = 1, 2, \ldots, n\).

The coefficients \(C_1, C_2, \ldots, C_n\) having been determined, the coefficients \(A_0, A_1, \ldots, A_{p-1}, B_0, B_1, \ldots, B_{q-1}\) can be determined by writing that identity (3.4.2) is satisfied when the function \(f\) is replaced by the polynomial

where \(h = 0, 1, \ldots, p + q - 1\). We have the system:

which determines the coefficients \(A_0, A_1, \ldots, A_{p-1}, B_0, B_1, \ldots B_{q-1}\), because the determinant of the system is the generalized Vandermonde determinant

Thus the problem posed for the quadrature formula (3.4.1) is completely solved.

Let us now determine the remainder \(R[f]\) of the quadrature formula (3.4.1) when the nodes and the coefficients have been determined by the problem posed above and when \(f \in C^{2n+p+q}[a, b]\). This problem was treated by us with the method of the function \(\varphi\), for an arbitrary weight \(\pi(x)\), integrable on the interval \([a, b]\) and positive on the interval \((a, b)\), in our paper [6].

But when \(\pi(x) = 1\), the determination of the remainder \(R[f]\) can be linked to the integral representation of a divided difference, as we shall see below.

3.4.2. Let us consider the divided difference of a function \(g\) on the nodes \(a, b\), of multiplicities \(p + 1\), \(q + 1\), and on the distinct double nodes \(x_1, x_2, \ldots, x_n\)

Proceeding as in § 3.2, we can write

Concerning this formula we have:

It suffices to give the proof for one of these coefficients, for example for \(P_1\).

Setting

we have

where \(V\) is the generalized Vandermonde determinant

We can also write

where \(L_1\) differs from \(V^{-1}\) by a factor of 1 or \(-1\).

Adding the elements of the last column to those of the first column, and then dividing the elements of the second row, of the third row, \(\ldots\), of the last row by \(1, 2, \ldots, N\), respectively, we can also write \(P_1\) in the form

In this form we see that the coefficient \(P_1\) is zero, because, \(x_1, x_2, \ldots, x_n\) being the roots of the polynomial \(\omega_n\), among the elements of the first \(n + p + q + 1\) columns there is one and the same linear combination, which results from formula (3.4.2) for \(f(x) = x^k\), where \(k = 0, 1, \ldots, N-1\).

Theorem 3.4.1 is thus proved.

3.4.3. This being so, let us replace in formula (3.4.15) the nodes \(x_1, x_2, \ldots, x_n\) by the roots of the polynomial \(\omega_n\), (3.4.7), determined by equations (3.4.8). Then, by Theorem 3.4.1, we have \(P_1 = 0, \ldots, P_n = 0\) and formula (3.4.15) becomes

If in this identity \(g(x)\) is replaced by 1, we deduce that

On the other hand, by Theorem 2.3.1 we have \(Q_0 \neq 0\), so that formula (3.4.20) can be written

Let us replace in this identity the function \(g\) by

and divide by \(Q_0\). We shall have the formula

which is precisely the formula (3.4.1) that we have in view and that we have regarded as an extension of the Gauss quadrature formula.

In this formula the nodes \(x_1, x_2, \ldots, x_n\) are the roots, real, distinct and lying between \(a\) and \(b\), of the polynomial \(\omega_n\), (3.4.6), given by equations (3.4.8), and the coefficients are related to the coefficients of the divided difference (3.4.20) by the formulas

The remainder \(R[f]\) is given by:

If we use the integral representation of the right-hand side of formula (3.4.25), we obtain the integral representation of the remainder \(R[f]\), when \(f \in C^N[a, b]\)

where the right-hand side contains the function \(\varphi\) from the integral representation of the divided difference (3.4.14) corresponding to the roots of the polynomial \(\omega_n\) determined by equations (3.4.8).

We thus have:

3.4.4. Replacing in formula (3.4.1) the function \(f\) by

where \(\omega_n\) is the polynomial determined by equations (3.4.7) and (3.4.8), which vanishes at \(x_1, x_2, \ldots, x_n\), and taking formula (3.4.26) into account, we obtain

But for the integral representation (3.4.27) we have \(\int_a^b \varphi\, ds = \frac{1}{(N+1)!}\) and therefore:

whence it follows that

which is an interesting interpretation of the coefficient \(\frac{1}{Q_0}\) on the right-hand side of formula (3.4.26).

§ 3.5. Extension of P. Turán's quadrature formula

3.5.1. In § 3.4 we considered an extension of the Gauss quadrature formula. Let us consider the quadrature formula of the form

as an extension of P. Turán's quadrature formula. In this formula \(\pi\) is a function integrable on the interval \([a, b]\) and positive on the interval \((a, b)\).

On the right-hand side there appear the derivatives \(f^{(l)}(x_j)\) of the function \(f\) for \(l = 0, 1, \ldots, 2(s-1)\) and \(j = 1, 2, \ldots, n\). The coefficients of formula (3.5.1) will be specified by the following fundamental problem, which we designate as:

We shall then have the identity

for \(f(x) = x^k\), where \(k = 0, 1, \ldots, 2ns + p + q - 1\).

To determine the nodes and the coefficients of formula (3.5.2) we must write the identities that result when the function \(f\) is replaced by

and

We can also write that formula (3.5.2) is verified when the function \(f\) is replaced by an arbitrary polynomial \(Q\) of degree at most \(p + q + 2sn - 1\).

Let us denote by

the polynomial whose roots are the nodes \(x_1, x_2, \ldots, x_n\) of identity (3.5.2). Determining the nodes \(x_1, x_2, \ldots, x_n\) is the same as determining the coefficients \(\lambda_1, \lambda_2, \ldots, \lambda_n\) of the polynomial \(\omega_n\).

The polynomials

where \(h = 0, 1, \ldots, n - 1\), are of degree \(p + q + (2s - 1)n + h \leq p + q + 2sn - 1\). The polynomial \(Q\) can also be written in the form

and we see that it has the roots \(a, b, x_1, \ldots, x_n\) of multiplicities \(p, q, 2s-1, \ldots, 2s-1\).

It follows that if in formula (3.5.2) the function \(f\) is replaced by the polynomial \(Q\), we must have

for \(h = 0, 1, \ldots, n-1\).

Equations (3.5.8) determine the polynomial \(\omega_n\) uniquely. By a theorem of D. Jackson [22], \(\omega_n\) is the minimizing polynomial of the integral

and it has been proved (Theorem 3.3.3) that the polynomial \(\omega_n\) has all its roots real, distinct and lying between \(a\) and \(b\).

Hence the nodes \(x_1, x_2, \ldots, x_n\) of formula (3.5.2) are the roots of the minimizing polynomial of the integral (3.5.9).

It remains to determine the coefficients of formula (3.5.2).

For this we must write that identity (3.5.2) is satisfied when the function \(f\) is replaced by the polynomials (3.5.3).

We shall have the linear equations

whose number is \((2s-1)n+p+q\), in \(p+(2s-1)n+q\) unknowns \(A_0, A_1, \ldots, A_{p-1}, C_j^{(0)}, C_j^{(1)}, \ldots, C_j^{2(s-1)}, j = 1, 2, \ldots, n\) and \(B_0, B_1, \ldots, B_{q-1}\).

This system determines all the coefficients because its determinant is the generalized Vandermonde determinant

Thus P. Turán's problem for the quadrature formula (3.5.1) is completely solved.

Once the nodes and the coefficients of formula (3.5.1) have been determined, it remains to determine the remainder \(R[f]\) of the formula, assuming that \(f \in C^{2ns+p+q}[a, b]\). This problem was treated by P. Pavel [28] with the method of the function \(\varphi\), for a weight \(\pi\) integrable on the interval \([a, b]\) and positive on the interval \((a, b)\).

But if \(\pi(x) = 1\), the determination of the remainder \(R[f]\) can be linked to the integral representation of a divided difference, as we shall see below.

3.5.2. Let us consider the divided difference of a function \(g\) on the nodes \(a, b\) of multiplicities \(p + 1, q + 1\) and on the distinct nodes \(x_1, x_2, \ldots, x_n\), each of multiplicity \(2s\)

Proceeding as in § 3.2, we can write

Concerning this formula we have:

It suffices to prove this theorem for one of the coefficients, for example for \(P_1^{(0)}\).

Setting

we have

where \(V\) is the generalized Vandermonde determinant

We can also write

where \(L_1\) differs from \(V^{-1}\) by a factor of 1 or \(-1\).

Adding the elements of the last column to the elements of the first column and then dividing the elements of the 2^nd, the 3^rd, \(\ldots\), the last row by \(1, 2, \ldots, N\), respectively, we can write \(P_1^{(0)}\) in the form

In this form we see that the coefficient \(P_1^{(0)}\) is zero, because, \(x_1, x_2, \ldots, x_n\) being the roots of the polynomial \(\omega_n\), among the elements of the first \((2s-1)n+p+q+1\) columns there is one and the same linear combination, which results from formula (3.5.2).

Theorem 3.5.1 is thus proved.

3.5.3. This being so, let us replace in formula (3.5.12) the nodes \(x_1, x_2, \ldots, x_n\) by the roots of the polynomial \(\omega_n\) determined by equations (3.5.5), (3.5.8). Then, by Theorem 3.5.1, we have \(P_1^{(0)} = 0, \ldots, P_n^{(0)} = 0\) and formula (3.5.12) becomes

If in this identity \(g(x)\) is replaced by 1, we have

and, by Theorem 2.3.1, we have \(Q_0 \neq 0\), so that formula (3.5.17) becomes

Let us replace in this identity the function \(g\) by

and divide by \(Q_0\). We shall have the formula

which is precisely the formula (3.5.1) that we have in view and that we regard as an extension of P. Turán's quadrature formula.

In formula (3.5.21) the nodes \(x_1, x_2, \ldots, x_n\) are the roots, real, distinct and lying between \(a\) and \(b\), of the polynomial \(\omega_n\) determined by equations (3.5.5), (3.5.8), which is the minimizing polynomial of the integral (3.5.9) corresponding to \(\pi(x) = 1\), and the coefficients are related to the coefficients of the divided difference (3.5.17) corresponding to the roots of the polynomial \(\omega_n\) by the formulas

The remainder \(R[f]\) is given by the expression

If we use the integral representation of the right-hand side of formula (3.5.23), we obtain the integral representation of the remainder \(R[f]\) when \(f \in C^N[a, b]\)

where the right-hand side contains the function \(\varphi\) from the integral representation of the divided difference (3.5.11) corresponding to the roots of the polynomial \(\omega_n\) determined by equations (3.5.5), (3.5.8).

We thus have

3.5.4. Replacing in formula (3.5.1), where \(\pi(x) = 1\), the function \(f\) by

where \(\omega_n\) is the polynomial determined by equations (3.5.5) and (3.5.8), and taking formula (3.5.24) into account, we have

But for the integral representation (3.5.25) we have \(\int_a^b \varphi\, dt = \frac{1}{(N+1)!}\) and therefore

whence it follows that

which is an interesting interpretation of the coefficient \(\frac{1}{Q_0}\) on the right-hand side of formula (3.5.24), related to the integral (3.5.9), whose minimizing polynomial is \(\omega_n\).

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In this chapter the method of the function \(\varphi\), set out in the first chapter in order to define the divided difference of a function of one variable and to obtain its integral representation, is extended to functions of two variables. We shall again arrive at this extension through a boundary value problem.

§ 4.1. The integration by parts formula for functions of two variables

In this section we establish the integration by parts formula for double integrals, and then the generalized integration by parts formula.

Formula (4.1.2) was called by Academician M. Nicolescu the integration by parts formula for double integrals [26].

We shall write formula (4.1.2) in the form

where the symbols on the right-hand side are the following:

To prove formula (4.1.2) we shall write its left-hand side in the form

and apply the integration by parts formula to the integral \(\int_{y_1}^{y_2}\). We shall have

Applying the integration by parts formula once more to each integral \(\int_{x_1}^{x_2}\), we obtain:

which is precisely formula (4.1.2).

We shall say that formula (4.1.6) is the generalized integration by parts formula for double integrals.

To prove formula (4.1.6), let us replace in formula (4.1.3) the function \(f\) by \(\dfrac{\partial^{2(n-1)} f}{\partial x^{n-1} \partial y^{n-1}}\). We shall have

Let us further replace in formula (4.1.3) the function \(\varphi\) by \(\dfrac{\partial^2 \varphi}{\partial x \partial y}\) and the function \(f\) by \(\dfrac{\partial^{2(n-2)} f}{\partial x^{n-2} \partial y^{n-2}}\). We shall have

We continue this procedure until we reach the formula

Adding all these formulas term by term, we obtain formula (4.1.6).

4.1.3. Let us consider the rectangle

the nodes \(M_{i,k}(x_i, y_k)\), where

and the rectangles

where \(i, k = 0, 1, \ldots, n-1\).

We also consider the functions \(\varphi_{i,k}\), \((i, k = 0, 1, \ldots, n-1)\), continuous on \(D_{i,k}\), together with the partial derivatives

Figure 4.1.1 shows the tiling of the rectangle \(D\) by the rectangles \(D_{i,k}\), with the nodes \(M_{i,k}\) marked.

To each rectangle \(D_{i,k}\) and to the functions \(f, \varphi_{i,k}\) we can apply the fundamental formula (4.1.6). Adding all these formulas term by term, we obtain a formula that is important for this chapter:

where

Taking formulas (4.1.4) into account, we can also write formula (4.1.10) in the form

The writing of formula (4.1.11) can be shortened by adopting the following convention:

With this convention, formula (4.1.11) can be written in the form

because \(\mathscr{P}[f]\) given by formula (4.1.12) coincides, in view of convention (A), with \(\mathscr{P}[f]\) given by formula (4.1.11).

§ 4.2. Boundary value problem

In connection with formula (4.1.12) or (4.1.11), we shall pose a boundary value problem with the aim of arriving at the definition of the divided difference of the function \(f\) on the nodes \(M_{i,k}\), \((i, k = 0, 1, \ldots, n)\), and at its representation by a double integral.

4.2.1. Formula (4.1.12) shows that \(\mathscr{P}[f]\) depends on \(\left( \dfrac{\partial^{2r} f}{\partial x^r \partial y^r} \right)_{x_i, y_k}\) for \(r = 0, 1, \ldots, n-1\) and \(i, k = 0, 1, \ldots, n\), on \(\left( \dfrac{\partial^{2r} f}{\partial x^r \partial y^r} \right)_{x, y_k}\) for \(r = 0, 1, \ldots, n-1\) and \(k = 0, 1, \ldots, n\), as well as on \(\left( \dfrac{\partial^{2r} f}{\partial x^r \partial y^r} \right)_{x_i, y}\) for \(r = 0, 1, \ldots, n-1\) and \(i = 0, 1, \ldots, n\).

In connection with formula (4.1.9) and with expression (4.1.12) of \(\mathscr{P}[f]\), it is of great interest to treat the following boundary value problem:

For this, the following boundary conditions must be introduced:

for \(r = 0, 1, \ldots, n-1\) and \(i, k = 0, 1, \ldots, n\). When we set \(i = n\), or \(k = n\), or \(i = k = 0\), or else \(i = k = n\) in the boundary conditions (4.2.2), we must take into account convention (A), made in subsection 4.1.3.

This boundary value problem will be denoted by \((P_{i,k})\), and the functions \(\varphi_{i,k}\) will be determined first on \(D_{0,0}\), then on \(D_{1,0}, \ldots, D_{n-1,0}\), next on \(D_{0,1}, \ldots, D_{n-1,1}, \ldots\), and so on.

The constants \(C_{i,k}\), \(i, k = 0, 1, \ldots, n\), will be determined, up to a factor, so that the problem is solvable. To determine them completely we shall introduce the supplementary condition

4.2.2. Problem \((P_{0,0})\). Setting \(i = k = 0\) in equations (4.2.1) and in the boundary conditions (4.2.2), and taking convention (A) into account, problem \((P_{0,0})\) consists in determining the solution of the partial differential equation

which satisfies the boundary conditions

This problem is very simple and has the solution

4.2.3. Problem \((P_{i,0})\), for \(i = 1, 2, \ldots, n-1\) and \(n\). Setting \(k = 0\) in equations (4.2.1) and in the boundary conditions (4.2.2), and taking convention (A) into account, problem \((P_{i,0})\) consists in determining the solution of the partial differential equation

which satisfies the boundary conditions:

for \(i = 1, 2, \ldots, n-1\) and \(n\), taking convention (A) into account. As can be seen, this is a recurrence problem. We shall first determine \(\varphi_{1,0}\) and then, by the method of complete induction, we shall determine \(\varphi_{i,0}\) for \(i = 1, 2, \ldots, n-1\).

4.2.4. Let us first integrate the partial differential equation (4.2.7) for \(i = 1\):

with the boundary conditions (4.2.8) corresponding to \(r = 0\), that is

We find

Then equation (4.2.9) is integrated with conditions (4.2.8) corresponding to \(r = 1\) and \(i = 1\), that is

We find

Let us suppose we have proved that

which holds for \(r = 1\) and \(r = 2\), and let us prove that the same formula is also valid for the index \(r+1\).

For this we must integrate equation (4.2.9ᵣ) with the boundary conditions (4.2.8) corresponding to the index \(r\):

Integrating, we find

Since formula (4.2.9ᵣ₊₁) is obtained from formula (4.2.9ᵣ) by replacing \(r\) by \(r+1\), it follows that formula (4.2.9ᵣ) is general; it holds for \(r = 1, 2, \ldots, n\).

For \(r = n\) we have

4.2.5. Let us suppose we have proved that

which is true for \(i = 1\), because in this case formula (4.2.13) reduces to formula (4.2.9ₙ). Let us prove that formula (4.2.13) is also valid for the index \(i + 1\).

For this we shall integrate the partial differential equation

first with the boundary conditions

and then with the boundary conditions

for \(r = 1, 2, \ldots, n-1\).

We first find

after that we find

and, applying the method of complete induction as in subsection 4.2.4, we find

which shows that formula (4.2.13) is general; it holds for \(i = 0, 1, \ldots, n-1\).

4.2.6. It remains to write that conditions (4.2.2) are satisfied for \(i = n\), \(k = 0\) and \(r = 0, 1, \ldots, n-1\). We must take convention (A) into account. These conditions reduce to:

and

for \(r = 0, 1, \ldots, n-1\).

Formula (4.2.13) gives, for \(i = n-1\),

For \(r = 0\), condition (4.2.17) is satisfied because \(\varphi_{n-1,0}\) is a polynomial of degree \(n-1\) in \(y\), whereas condition (4.2.18) leads to the relation

For \(r = 1, 2, \ldots, n-1\), conditions (4.2.17) are

and for them to be satisfied we must take

Conditions (4.2.18) are satisfied, because the derivatives \(\frac{\partial^{2(n-r-1)}\varphi_{n-1,0}}{\partial x^{n-r-1}\partial y^{n-r-1}}\) have the property of vanishing when \(x = x_n\), for \(r = 1, 2, \ldots, n-1\).

Indeed, we have

and the large bracket is zero by equations (4.2.21) for \(r = 1, 2, \ldots, n-1\).

Relations (4.2.20) and (4.2.21) can be written in the form:

and these equations determine \(C_{0,0}, C_{1,0}, \ldots, C_{n,0}\) up to a factor. We have

and therefore

The factor \(H_0\) will be determined later.

4.2.7. Problem \((P_{0,k})\) for \(k = 1, 2, \ldots, n-1\) and \(n\). Setting \(i = 0\) in equation (4.2.1) and in the boundary conditions (4.2.2), and taking convention (A) into account, problem \((P_{0,k})\) consists in determining the solution of the partial differential equation:

which satisfies the boundary conditions:

for \(k = 1, 2, \ldots, n-1\) and \(n\), taking convention (A) into account.

The solution of this problem is found as in the case of problem \((P_{i,0})\). We shall have

where the coefficients \(C_{0,0}, C_{0,1}, \ldots, C_{0,n}\) are given by the equations

and

for \(r = 1, 2, \ldots, n-1\).

The system formed by equations (4.2.27), (4.2.28) can also be written in the form:

4.2.8. Problem \((P_{ik})\), \(i, k = 1, 2, \ldots, n-1\) and \(n\). This problem was stated in subsection 4.2.1. It is a recurrence problem with two indices \(i\) and \(k\). Setting first \(k = 1\), we shall determine \(\varphi_{1,1}, \varphi_{2,1}, \ldots, \varphi_{n-1,1}\). Then, by the method of complete induction, we shall determine, for the index \(k\), the functions \(\varphi_{1,k}, \varphi_{2,k}, \ldots, \varphi_{n-1,k}\). Taking into account convention (A) and condition (4.2.3), all the coefficients \(C_{i,k}\) will be determined.

Let us begin by determining \(\varphi_{1,1}\). We have to integrate the partial differential equation

with the boundary conditions

for \(r = 0, 1, \ldots, n-1\).

The functions \(\varphi_{0,0}, \varphi_{1,0}, \varphi_{0,1}\) being determined by equations (4.2.6), (4.2.13), (4.2.26), we can deduce from formulas (4.2.31) the boundary conditions for the function \(\varphi_{1,1}\) and thus determine \(\varphi_{1,1}\).

We have

so that conditions (4.2.31) become, for \(r = 0\),

we find

For \(r = 1, 2, \ldots, n-1\), the boundary conditions are:

Integrating equation (4.2.32) with conditions (4.2.31₁), we find

Let us suppose we have proved that

which is true for \(r = 2\), because in this case formula (4.2.34) coincides with formula (4.2.33). Let us prove that formula (4.2.34) is also valid for the index \(r + 1\). For this we must integrate equation (4.2.34) with conditions (4.2.31ᵣ).

Proceeding as above, we find

Since this formula is obtained from formula (4.2.34) by replacing \(r\) by \(r + 1\), it follows that formula (4.2.34) is general; it holds for \(r = 1, 2, \ldots, n\). For \(r = n\), formula (4.2.34) is

4.2.9. Let us suppose it has been proved that

which is true for \(i = 1\), because in this case formula (4.2.37) coincides with formula (4.2.36). Let us prove that formula (4.2.37) is also valid for the index \(i + 1\).

For this we must integrate the partial differential equation

with the boundary conditions

To these conditions we further add formulas (4.2.37) and (4.2.13) for the indices \(i\) and \(i + 1\), which we have already proved.

Let us first integrate equation (4.2.38) with conditions (4.2.39₀), which are the following:

We find

For \(r = 1\), formulas (4.2.39₁) give

Integrating equation (4.2.40) with these conditions, we find

Let us suppose it has been proved that

which is true for \(r = 2\), because, in this case, formula (4.2.42) coincides with formula (4.2.41). Let us prove that formula (4.2.42) is also valid for the index \(r + 1\).

For this we must integrate equation (4.2.42) with conditions (4.2.39ᵣ). Taking into account formulas (4.2.37) and (4.2.13) for the indices \(i\) and \(i + 1\), the boundary conditions give

Integrating equation (4.2.42) with these conditions, we find

Formula (4.2.43) is obtained from formula (4.2.42) by replacing \(r\) by \(r + 1\). Formula (4.2.42) is therefore general; it holds for \(r = 1, 2, \ldots, n\). For \(r = n\) we have

Formula (4.2.44) is obtained from formula (4.2.37) by replacing \(i\) by \(i + 1\). Hence formula (4.2.37) is general; it holds for \(i = 1, 2, \ldots, n-1\).

4.2.10. It remains to write that conditions (4.2.2) are satisfied for \(i = n\), \(k = 1\) and \(r = 0, 1, \ldots, n-1\). We must take convention (A) into account. These conditions reduce to:

We must take into account formula (4.2.19)

For \(r = 0\), condition (4.2.45) is satisfied, because \(\varphi_{n-1,1}\) is a polynomial of degree \(n-1\) in \(y\). Condition (4.2.46) reduces to:

For \(r = 1, 2, \ldots, n-1\) we have

But by equations (4.2.21) the first large bracket on the right-hand side is zero for \(r = 1, 2, \ldots, n-1\). There remains

For conditions (4.2.45) to be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

for \(r = 1, 2, \ldots, n-1\).

On the other hand, we have

and it follows from equations (4.2.48) that conditions (4.2.46) are satisfied for \(r = 1, 2, \ldots, n-1\).

Hence the coefficients \(C_{0,1}, \ldots, C_{n,1}\) are determined by equations (4.2.47) and (4.2.48). The system formed by these equations can also be written in the form

We have

where \(H_1\) is a factor that will be determined later. From these equations we deduce

4.2.11. Let us suppose it has been proved that

for \(j = 1, 2, \ldots, k\) and \(i = 1, 2, \ldots, n-1\), where the coefficients \(C_{i,j}\) of this formula are given by the equations:

and

for \(j = 0, 1, \ldots, k\) and \(r = 1, 2, \ldots, n-1\). The system formed by these equations can also be written in the form:

and gives the following relations:

where the coefficient \(H_j\) will be determined later on. It follows that

for \(i = 0, 1, \ldots, n\).

Formula (4.2.51) is true for \(j = 1\), since in this case it coincides with formula (4.2.37). Likewise, the system formed by equations (4.2.52) and (4.2.53) is true for \(j = 0, 1\), since in this case it coincides with equations (4.2.20), (4.2.21) and (4.2.47), (4.2.48).

We shall show further that formula (4.2.51) is valid for \(j = k + 1\) and that the coefficients \(C_{0,k+1}, C_{1,k+1}, \ldots, C_{n,k+1}\) are also given by equations (4.2.52), (4.2.53) for \(j = k + 1\).

4.2.12. In order to prove that formula (4.2.51) is valid for \(j = k + 1\), we must first integrate the partial differential equation

with the boundary conditions

for \(r = 0, 1, \ldots, n-1\).

By formula (4.2.51), and for \(i = 1\) and \(j = k\), we have

and then, by formula (4.2.26), we have

and

Proceeding as in subsection 4.2.8, one shows that for \(r = 0\) we have

Further, for \(r = 1\) we have

and, in general, for the index \(r\) we have

This formula is valid for \(r = 1, 2, \ldots, n\) and therefore, for \(r = n\), we have

4.2.13. Let us suppose it has been proved that

which is true for \(i = 1\), since in this case formula (4.2.62) coincides with formula (4.2.61). We shall show that formula (4.2.62) is also valid when the index \(i\) is replaced by \(i + 1\).

To this end we shall integrate the partial differential equation

with the boundary conditions (4.2.2) corresponding to the indices \(i + 1\), \(k + 1\), that is

for \(r = 0, 1, \ldots, n-1\).

We must take into account formula (4.2.51) corresponding to \(j = k\), that is

then the same formula, but with the index \(i\) replaced by \(i + 1\), that is

and finally formula (4.2.62).

Proceeding as in subsection 4.2.8, one shows that for \(r = 0\) we have

Further, for \(r = 1\) we have

and, in general, for the index \(r\), we have

This formula is valid for \(r = 1, 2, \ldots, n\). For \(r = n\) we have

which proves that formula (4.2.62) is valid when the index \(i\) is replaced by \(i + 1\). Formula (4.2.62) is therefore valid for \(i = 1, 2, \ldots, n-1\).

4.2.14. It remains to write that conditions (4.2.2) are satisfied for \(\varphi_{i,k+1}\) when \(i = n\) and \(r = 0, 1, \ldots, n-1\). Taking into account convention (A), these conditions are:

Formula (4.2.62) for \(i = n-1\) becomes

while formula (4.2.51) for \(i = n-1\) and \(j = k\) is

Let us consider conditions (4.2.71) and (4.2.72) for \(r = 0\); they are

The first condition is satisfied, \(\varphi_{n-1,k+1}(x_n, y)\) being a polynomial in \(y\) of degree \(n-1\). The second condition leads to the equation

Let us consider conditions (4.2.71) and (4.2.72) for \(r = 1, 2, \ldots, n-1\). For conditions (4.2.71) we have

Equations (4.2.53) are satisfied for \(j = 0, 1, \ldots, k\) and \(r = 1, 2, \ldots, n-1\). The first \(k + 1\) brackets on the right-hand side are therefore zero. There remains

and in order that conditions (4.2.71) be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

On the other hand, we have

and, by equations (4.2.76), conditions (4.2.72) are satisfied for \(r = 1, 2, \ldots, n-1\).

We have thus proved that the system formed by equations (4.2.52) and (4.2.53) is also true for \(j = k + 1\); it is then true for \(j = 0, 1, \ldots, n-1\), and formulas (4.2.55) are valid for \(i = 0, 1, \ldots, n\) and \(j = 0, 1, \ldots, n-1\).

4.2.15. We must examine the boundary conditions for \(\varphi_{i,k}\) in the case when \(i = 0, 1, \ldots, n-1\) and \(k = n\). It was already shown in subsection 4.2.7, when problem \((P_{0,k})\) was treated, that the coefficients \(C_{0,0}, C_{0,1}, \ldots, C_{0,n}\) are given by the system of equations

It remains to show that for \(i = 1, 2, \ldots, n-1\) and \(r = 1, 2, \ldots, n-1\) we have

To this end let us set \(k = n\) in conditions (4.2.2) and take into account convention (A). We shall have the conditions:

and

for \(i = 1, 2, \ldots, n-1\) and \(r = 0, 1, \ldots, n-1\).

We may also write \(\varphi_{i,n-1}\), given by formula (4.2.65) for \(k = n-1\), in the form

and we shall also consider the same formula when the index \(i\) is replaced by \(i-1\).

We first observe that condition (4.2.79) for \(r = 0\) is satisfied, since \(\varphi_{i,n-1}(x, y_n)\) is a polynomial of degree \(n-1\) in \(x\).

Then condition (4.2.80) for \(r = 0\) leads to the equation

Let us now examine conditions (4.2.79) and (4.2.80) for \(r = 1, 2, \ldots, n-1\). We have

For \(i = 1\), this formula reduces to:

since, by equations (4.2.28), the first large bracket on the right-hand side is zero for \(r = 1, 2, \ldots, n-1\).

In order that conditions (4.2.79) be satisfied when \(r = 1, 2, \ldots, n-1\), we must take

It is shown in the same way that conditions (4.2.79) for \(i = 2, 3, \ldots, n-1\) and \(r = 1, 2, \ldots, n-1\) lead to equations (4.2.78).

On the other hand, we have

and from equations (4.2.78) it follows that conditions (4.2.80) are satisfied for \(r = 1, 2, \ldots, n-1\).

Equations (4.2.77) and (4.2.78) lead to the system

valid for \(i = 0, 1, \ldots, n-1\).

4.2.16. The function \(\varphi_{n-1,0}\) has been determined by formula (4.2.19):

where the constants \(C_{0,0}, C_{1,0}, \ldots, C_{n,0}\) are determined by the equations

The function \(\varphi_{n-1,0}\) can also be written in the form

Indeed, we have

Taking into account equations (4.2.22), the preceding formula becomes

which means that formula (4.2.87) is proved.

The function \(\varphi_{n-1,1}\) can also be written in the form

Indeed, formula (4.2.37), for \(i = n-1\), is

But since the constants \(C_{0,0}, C_{1,0}, \ldots, C_{n,0}\) and \(C_{0,1}, C_{1,1}, \ldots, C_{n,1}\) are connected by equations (4.2.21) and

one shows as above that

whence formula (4.2.88) follows.

In general, the function \(\varphi_{n-1,k}\) can be written in the form

For \(k = n-1\), we shall need the formula

4.2.17. The function \(\varphi_{0,n-1}\) has been determined by formula (4.2.26), corresponding to \(k = n-1\),

where the constants \(C_{0,0}\), \(C_{0,1}\), \(\ldots, C_{0,n}\) are determined by the equations

The function \(\varphi_{0,n-1}\) can also be written in the form

Indeed, with the help of equations (4.2.29), one shows as in subsection 4.2.16 that

and therefore we have formula (4.2.92).

The function \(\varphi_{i,n-1}\) can also be written in the form:

Indeed, one takes into account formula (4.2.65) for \(k = n-1\),

and one shows as in subsection 4.2.16, with the help of the systems of equations (4.2.86) for \(i = 0, 1, \ldots, n-1\), that the large brackets on the right-hand side are equal to \(-C_{0,n}\frac{(y-y_n)^{n-1}}{(n-1)!}, \ldots, -C_{i,n}\frac{(y-y_n)^{n-1}}{(n-1)!}\). Formula (4.2.93) then follows immediately.

For \(i = n-1\), we shall need the formula

4.2.18. Let us now determine the constant \(C_{n,n}\) and show that the system of equations (4.2.54), which is valid for \(j = 0, 1, \ldots, n-1\), is also valid for \(j = n\), and that the system of equations (4.2.86), which is valid for \(i = 0, 1, \ldots, n-1\), is also valid for \(i = n\).

Let us write the boundary conditions (4.2.2) corresponding to \(i = n\), \(k = n\). Taking into account convention (A), we shall have the conditions

for \(r = 0, 1, \ldots, n-1\).

If we use formula (4.2.91), condition (4.2.95) for \(r = 0\) gives the equation

and if we use formula (4.2.94), condition (4.2.95) for \(r = 0\) gives the equation

For \(r = 1, 2, \ldots, n-1\), conditions (4.2.95) are satisfied, since by formula (4.2.91) or (4.2.94) we have

and for \(x = x_n\) and \(y = y_n\) the right-hand side is zero for \(r = 1, 2, \ldots, n-1\).

Let us now consider conditions (4.2.2) for \(i = n\), \(k = n-1\) and then for \(i = n-1\), \(k = n\). From these conditions we take only

For \(r = 0\) these conditions are satisfied, so that we may consider them only for \(r = 1, 2, \ldots, n-1\).

Using formula (4.2.91) we have

In order that conditions (4.2.99) be satisfied, we must take

Using formula (4.2.94) we shall have

In order that conditions (4.2.98) be satisfied, we must take

Thus the constants \(C_{n,0}, C_{n,1}, \ldots, C_{n,n}\) are determined by the system formed by equations (4.2.96) and (4.2.100), that is, by the system

and the constants \(C_{0,n}, C_{1,n}, \ldots, C_{n,n}\) are determined by the system formed by equations (4.2.97) and (4.2.101), that is, by the system

Hence the system of equations (4.2.54) is also valid for \(j = n\), and the system of equations (4.2.86) is also valid for \(i = n\).

A consequence of this fact is that the function \(\varphi_{n-1,n-1}\) can also be written in the form

Indeed, let us consider formula (4.2.91), which gives the function \(\varphi_{n-1,n-1}\). One shows as in subsection 4.2.16 that, by equations (4.2.103), we have

and therefore we have formula (4.2.104).

4.2.19. To sum up, we have on the one hand the system of equations

which is satisfied by the constants \(C_{0,k}, C_{1,k}, \ldots, C_{n,k}\), and this for \(k = 0, 1, \ldots, n\), and on the other hand the system of equations

which is satisfied by the constants \(C_{i,0}, C_{i,1}, \ldots, C_{i,n}\), and this for \(i = 0, 1, \ldots, n\).

Equations (4.2.105) show that

and therefore

where the \(H_k\) are constants.

Replacing in equations (4.2.106) the constants \(C_{ik}\) by formulas (4.2.107) for \(k = 0, 1, \ldots, n\), the index \(i\) being fixed, we obtain the equations

which give

where \(H\) is a constant which we shall determine by condition (4.2.3).

The preceding equations show that

and therefore

4.2.20. Replacing, in the expression of \(\mathscr{P}[f]\) given by formula (4.1.12), the functions \(\varphi_{i,k}\) by the solution of the boundary value problem, we have

and, using formulas (4.2.110), we can write

Condition (4.2.3) requires that

Replacing in formula (4.2.112) the function \(f(x, y)\) by \(f(x, y) = x^n y^n\), we shall have the equation

which determines the constant \(H\).

However,

so that equation (4.2.113) becomes \(1 = H V(x_0, x_1, \ldots, x_n) V(y_0, y_1, \ldots, y_n)\) and therefore

Formulas (4.2.110) then give

for \(i, k = 0, 1, \ldots, n\).

The constants \(C_{i,k}\) having been determined, the boundary value problem is completely solved.

§ 4.3. The divided difference of order \(n\) of the function \(f\) on a grid of nodes \(M_{i,k}\), \((i, k = 0, 1, \ldots, n)\) and its integral representation

4.3.1. When in formula (4.1.12) the functions \(\varphi_{i,k}\) are replaced by the solution of the boundary value problem, the functional \(\mathscr{P}[f]\) becomes

where the coefficients \(C_{i,k}\) are given by formulas (4.2.115):

for \(i, k = 0, 1, \ldots, n\).

Formula (4.3.4) is identical with formula (4.3.1).

The proof of this theorem is very simple. It suffices to replace in formula (4.1.9) the functions \(\varphi_{i,k}\) by the solution of the boundary value problem treated in § 4.2, and we shall have

which means that we have the integral representation (4.3.6), where the function \(\varphi\) coincides on each rectangle \(D_{ik}\) with the function \(\varphi_{i,k}\).

4.3.3. Connection between the divided difference of a function of two variables and the divided differences of functions of one variable.

It is known that if one considers the divided difference of the function \(u\) on the nodes \(x_0, x_1, \ldots, x_n\), then we have

where

and likewise, if one considers the divided difference of the function \(v\) on the nodes \(y_0, y_1, \ldots, y_n\), then we have

where

To prove this theorem it suffices to take into account formulas (4.3.8), (4.3.10) and (4.3.2).

Returning to formula (4.3.4) and replacing the coefficients \(C_{i,k}\) by the product \(\lambda_i \mu_k\), we can write

Taking into account formulas (4.3.9) and (4.3.7), we shall have

that is

This is the formula which T. Popoviciu [31] took a priori as the definition of the divided difference of a function of two variables.

It is evident that one can also write

4.3.4. If \(u \in C^n[x_0, x_n]\), the divided difference of order \(n\) of the function \(u\) on the nodes \(x_0, x_1, \ldots, x_n\), given by formula (4.3.7), has an integral representation

where \(\psi\) is the spline function:

which coincides on the intervals \([x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]\) with the polynomials

Likewise, if \(v \in C^n[y_0, y_n]\), the divided difference of order \(n\) of the function \(v\) on the nodes \(y_0, y_1, \ldots, y_n\), given by formula (4.3.9), has an integral representation

where \(\theta\) is the spline function

which coincides on the intervals \([y_0, y_1], [y_1, y_2], \ldots, [y_{n-1}, y_n]\) with the polynomials

In the integral representation (4.3.6) the function \(\varphi\) coincides on each rectangle \(D_{i,k}\) with the function \(\varphi_{i,k}\) given by formula (4.2.65)

Replacing in this formula the constants \(C_{i,k}\) by formulas (4.3.11), for \(i = 0, 1, \ldots, i\) and \(k = 0, 1, \ldots, k\), the preceding formula can be written in the form

Taking into account formulas (4.3.16) and (4.3.19), we can write the preceding formula in the form

and therefore we have the following theorem:

Indeed, it suffices to observe that this formula is true on each rectangle \(D_{i,k}\), as formula (4.3.20) shows.

Indeed, it has been shown that the spline functions \(\psi\) and \(\theta\) are positive, the first on the interval \((x_0, x_n)\), the second on the interval \((y_0, y_n)\). It then follows from formula (4.3.21) that the function \(\varphi\) is positive on the open rectangle \(D\) and vanishes on the boundary of this rectangle.

The function \(\varphi\) is an example of a spline function of two variables.

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Română

In this chapter the method of the function \(\varphi\) is extended to functions of three variables, with the aim of arriving at the definition of the divided difference of order \(n\) of a function of three variables on a grid \(M_{i,j,k}\), \((i, j, k = 0, 1, \ldots, n)\), and at its representation by a triple integral. We shall again make this extension by means of a boundary value problem [20].

§ 5.1. The integration by parts formula for functions of three variables

In this section we shall establish, as in § 4.1, the integration by parts formula for the triple integral, and then the generalized integration by parts formula.

5.1.1. We recall here the integration by parts formula for the functions \(f\) and \(\varphi\), continuous on the interval \([x_1, x_2]\) together with their derivatives \(f'\) and \(\varphi'\). This formula is

Next we consider the functions \(f\) and \(\varphi\) of two variables \(x\) and \(y\), continuous together with their partial derivatives

on a rectangle defined by the inequalities \(x_1 \leq x \leq x_2\), \(y_1 \leq y \leq y_2\).

For these functions we established in § 4.1 the integration by parts formula for double integrals:

The proof and the notation of this formula were given in § 4.1.

Indeed, by the integration by parts formula (5.1.1), we can write

Applying the integration by parts formula for double integrals (5.1.3) to the first two double integrals and to the double integral \(\int_{x_1}^{x_2} \int_{y_1}^{y_2} \frac{\partial \varphi}{\partial z} \frac{\partial^2 f}{\partial x \partial y}\, dx\, dy\), we shall have

that is

which means that we have formula (5.1.5).

5.1.3. The method we have given for obtaining the integration by parts formula (5.1.5) for a triple integral extends to multiple integrals in general. We shall not, however, dwell on these extensions, and we pass to applications.

5.1.4. The generalized integration by parts formula for triple integrals. Let us suppose that the functions \(f\) and \(\varphi\) of three variables \(x, y, z\) are continuous on the parallelepiped \(\Delta\) together with the partial derivatives:

for \(p = 1, 2, \ldots, n\).

Under this hypothesis we may replace in formula (5.1.5) the partial derivative \(\dfrac{\partial^3 f}{\partial x \partial y \partial z}\) by \(\dfrac{\partial^{3n} f}{\partial x^n \partial y^n \partial z^n}\). We shall have

To the last integral one may again apply the integration by parts formula, replacing the function \(\varphi\) by \(\dfrac{\partial^3 \varphi}{\partial x \partial y \partial z}\) and the function \(\dfrac{\partial^3 f}{\partial x \partial y \partial z}\) by \(\dfrac{\partial^{3(n-1)} f}{\partial x^{n-1} \partial y^{n-1} \partial z^{n-1}}\). Further, one may apply the integration by parts formula to the new triple integral, replacing the function \(\varphi\) by \(\dfrac{\partial^6 \varphi}{\partial x^2 \partial y^2 \partial z^2}\) and \(\dfrac{\partial^{3(n-2)} f}{\partial x^{n-2} \partial y^{n-2} \partial z^{n-2}}\), \(\ldots\), and so on.

One finally arrives at the generalized integration by parts formula for a triple integral

We have thus obtained:

5.1.5. Let us consider the parallelepiped \(D\) defined by the inequalities

and the grid of nodes \(M_{i,j,k}\), \((i, j, k = 0, 1, \ldots, n)\), with coordinates \((x_i, y_j, z_k)\), where

We shall denote by \(D_{i,j,k}\) the parallelepiped defined by the inequalities

where \(i, j, k = 0, 1, \ldots, n-1\).

To each parallelepiped \(D_{i,j,k}\) we associate a function \(\varphi_{i,j,k}\), continuous on \(D_{i,j,k}\) together with its partial derivatives of the same orders as the partial derivatives appearing in the sequence (5.1.6). Likewise, to the parallelepiped \(D\) we associate a function \(f\), continuous on \(D\) together with its partial derivatives of the same orders as the partial derivatives appearing in the sequence (5.1.6). Then the generalized integration by parts formula for triple integrals (5.1.7) or (5.1.8) can be applied to each parallelepiped \(D_{i,j,k}\) and to the functions \(f, \varphi_{i,j,k}\). Adding all these formulas term by term, we shall have the formula

where we have denoted by \(\mathscr{P}[f]\) the functional

When the node \(M_{i,j,k}\) is in the interior of the parallelepiped \(D\), the coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) results from formula (5.1.8) applied to all the parallelepipeds \(D_{i,j,k}\) having one of their vertices at the point \(M_{i,j,k}\). These parallelepipeds are:

Taking into account formula (5.1.8), the coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) in \(\mathscr{P}[f]\) is

Let us consider the node \(M_{i,j,0}\) situated on one of the faces of the parallelepiped \(D\), which means that \(0 \lt i, j \lt n\) and \(k = 0\). Then there are four parallelepipeds \(D_{i,j,0}\); \(D_{i-1,j-1,0}\); \(D_{i,j-1,0}\); \(D_{i-1,j,0}\) having the node \(M_{i,j,0}\) as a vertex.

The coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_0)\) in \(\mathscr{P}[f]\) is

The coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_0, z_k)\) in \(\mathscr{P}[f]\) is

The coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_0, y_j, z_k)\) in \(\mathscr{P}[f]\) is

Formulas (5.1.13), (5.1.14), (5.1.15) are written for \(0 \lt i, j \lt n\); \(0 \lt i, k \lt n\); \(0 \lt j, k \lt n\).

Likewise, the coefficients of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_n)\), \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_n, z_k)\), \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_n, y_j, z_k)\) in \(\mathscr{P}[f]\) are

Formulas (5.1.16), (5.1.17), (5.1.18) are written for \(0 \lt i, j \lt n\); \(0 \lt i, k \lt n\); \(0 \lt j, k \lt n\).

The coefficient (5.1.12) of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) was written for nodes interior to the parallelepiped \(D\), the indices \(i, j, k\) not taking the values 0 or \(n\). The coefficients (5.1.13)—(5.1.18) can be made to enter into the general coefficient by adopting:

Indeed, taking this convention into account, the coefficient (5.1.12) reduces to the coefficient (5.1.13) when \(k = 0\), and the coefficient (5.1.12) reduces to the coefficient (5.1.16) when \(k = n, \ldots\).

For the nodes

situated on the edges of the parallelepiped \(D\), the coefficients of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}\) in \(\mathscr{P}[f]\) are the following:

By convention (A), the coefficients (5.1.20) are deduced from the general coefficient by setting \(j = 0, k = 0, \ldots, j = n, k = n\).

For the vertices of the parallelepiped \(D\):

the coefficients of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) are:

By convention (A), the coefficients (5.1.22) are deduced from the general coefficient by setting \(i = 0, j = 0, k = 0, \ldots, i = n, j = n, k = n\).

Hence, writing in \(\mathscr{P}[f]\) the general term

one obtains all the terms of this form by letting the indices \(i, j, k\) take the values from 0 to \(n\) and taking convention (A) into account.

5.1.6. The coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) under the integral sign \(\int_{x_i}^{x_{i+1}}\) in \(\mathscr{P}[f]\), when \(0 \lt j, k \lt n\), results from formula (5.1.8) applied to all the parallelepipeds \(D_{i,j,k}\) which have the edge \([x_i, x_{i+1}]\) in common; these parallelepipeds are:

The contribution of these parallelepipeds to the coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y_j, z_k)\) is

We shall say that (5.1.24) is the general term of this form.

If \(k = 0\) but \(0 \lt j \lt n\), the parallelepipeds \(D_{i,j,0}\), \(D_{i,j-1,0}\) have the edge \([x_i, x_{i+1}]\) in common, and in \(\mathscr{P}[f]\) we have:

If \(k = n\) but \(0 \lt j \lt n\), only the parallelepipeds \(D_{i,j-1,n-1}\), \(D_{i,j,n-1}\) have the edge \([x_i, x_{i+1}]\) in common, and in \(\mathscr{P}[f]\) we have:

Likewise, if \(j = 0\) or \(j = n\), but \(0 \lt k \lt n\), only the parallelepipeds \(D_{i,0,k-1}\), \(D_{i,0,k}\) or \(D_{i,n-1,k-1}\), \(D_{i,n-1,k}\) have the edge \([x_i, x_{i+1}]\) in common, and in \(\mathscr{P}[f]\) we have:

or

Finally, if \(j = 0\), \(k = 0\), only the parallelepiped \(D_{i,0,0}\) has the edge \([x_i, x_{i+1}]\), and in \(\mathscr{P}[f]\) we have

Likewise, if \(j = n\), \(k = 0\), or \(j = 0\), \(k = n\), or \(j = n\), \(k = n\), the corresponding terms, analogous to the term (5.1.29), are

It should be observed that the terms (5.1.25)—(5.1.32) are deduced from the general term (5.1.24) by means of convention (A), made about the function \(\varphi_{i,j,k}\), when at least one of the indices is equal to 0 or to \(n\).

The same remarks apply to the integrals \(\int_{y_j}^{y_{j+1}}\) and \(\int_{z_k}^{z_{k+1}}\) in \(\mathscr{P}[f]\).

For the integrals \(\int_{y_j}^{y_{j+1}}\) in \(\mathscr{P}[f]\) we have, in the general case \(0 \lt i, k \lt n\), the general term

and all the other terms obtained from the general term when at least one of the indices \(i\), \(k\) has the value 0 or \(n\), taking convention (A) into account.

Likewise, for the integrals \(\int_{z_k}^{z_{k+1}}\) in \(\mathscr{P}[f]\) we have, in the general case \(0 \lt i, j \lt n\), the general term

and all the other terms obtained from the general term when at least one of the indices \(i\), \(j\) has the value 0 or \(n\), taking convention (A) into account.

5.1.7. The coefficient of \(\dfrac{\partial^{3r} f}{\partial x^r \partial y^r \partial z^r}(x_i, y, z)\) in the double integral \(\int_{y_j}^{y_{j+1}} \int_{z_k}^{z_{k+1}}\) in \(\mathscr{P}[f]\), when \(0 \lt i \lt n\), results from formula (5.1.8) applied to the parallelepipeds

We thus have the general term

For \(i = 0\) and \(i = n\) we have the terms:

These terms are deduced from the general term (5.1.36) by setting \(i = 0\) and \(i = n\) and taking convention (A) into account.

For the double integral \(\int_{x_i}^{x_{i+1}} \int_{z_k}^{z_{k+1}}\) in \(\mathscr{P}[f]\) we have, for \(0 \lt j \lt n\), the general term

and the terms obtained from the general term for \(j = 0\) and \(j = n\), taking convention (A) into account.

Likewise, for the double integral \(\int_{x_i}^{x_{i+1}} \int_{y_j}^{y_{j+1}}\) in \(\mathscr{P}[f]\) we have, for \(0 \lt k \lt n\), the general term

and the terms obtained from the general term for \(k = 0\) and \(k = n\), taking convention (A) into account.

5.1.8. Gathering the results of the discussions carried out in subsections 5.1.5, 5.1.6, 5.1.7, we can write formula (5.1.11) in the following form:

In applying this formula, convention (A) must be taken into account, and in this way all the terms of formula (5.1.11) are obtained.

§ 5.2. Boundary value problem

Formulas (5.1.10) and (5.1.11), or (5.1.41), of § 5.1 are important, as we shall see in the following sections. In connection with formula (5.1.41) we shall pose a boundary value problem with the aim of arriving at the definition of the divided difference of order \(n\) of the function \(f\) on the nodes \(M_{i,j,k}\), where \(i, j, k = 0, 1, \ldots, n\), and at its integral representation. This problem is lengthy to treat, and for this reason it will be the subject of several sections.

5.2.1. Formula (5.1.41) shows that \(\mathscr{P}[f]\) depends on:

for \(r = 0, 1, \ldots, n-1\) and \(i, j, k = 0, 1, \ldots, n\).

It is then of great interest to treat the following boundary value problem:

For this purpose boundary conditions must be introduced, by requiring that the coefficients of the partial derivatives (5.2.3), (5.2.2) in formula (5.1.41), in \(\int_{y_j}^{y_{j+1}} \int_{z_k}^{z_{k+1}}, \ldots\) and in \(\int_{x_i}^{x_{i+1}}, \ldots\), be zero, and likewise that the coefficients of the partial derivatives (5.2.1) be zero for \(r = 1, 2, \ldots, n-1\), while they reduce to constants for \(r = 0\).

The boundary conditions that we shall introduce will therefore be the following:

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Then

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Finally

for \(i, j, k = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

When one of the indices in \(\varphi_{i,j,k}\) is replaced by \(-1\) or \(n\), convention (A), made in subsection 5.1.5, must be taken into account.

The boundary value problem for the determination of the function \(\varphi_{i,j,k}\) on the parallelepiped \(D_{i,j,k}\) will be denoted by \((P_{i,j,k})\).

In general, the solution of problem \((P_{i,j,k})\) depends on the solutions of problems \((P_{i-1,j-1,k})\), \((P_{i,j-1,k})\), \((P_{i-1,j,k})\), \((P_{i,j,k-1})\), \((P_{i-1,j-1,k-1})\), \((P_{i,j-1,k-1})\), \((P_{i-1,j,k-1})\), so that at first sight it appears to be a difficult problem. However, by proceeding methodically, we shall succeed in solving it.

We shall determine systematically \(\varphi_{0,0,0}\), \(\varphi_{i,0,0}\), \(\varphi_{0,j,0}\), \(\varphi_{0,0,k}\) for \(i, j, k = 1, 2, \ldots, n-1\). Then we shall determine \(\varphi_{i,j,0}\), \(\varphi_{0,j,k}\), \(\varphi_{i,0,k}\) for \(i, j, k = 1, 2, \ldots, n-1\). Finally, we shall determine \(\varphi_{i,j,k}\) for \(i, j, k = 1, 2, \ldots, n-1\).

The constants \(C_{i,j,k}\), where \(i, j, k = 0, 1, \ldots, n\), will be determined so that the boundary value problem is possible. They will not be determined completely by the preceding boundary conditions, and will depend on a factor. In order to determine this factor, a new condition will be imposed

5.2.2. If in conditions (5.2.5) and (5.2.6) we set \(i = 0\), \(j = 0\), \(k = 0\) and take convention (A) into account, we have:

The solution of this problem is

5.2.3. If in conditions (5.2.5) and (5.2.6) we set \(j = 0\), \(k = 0\) and take convention (A) into account, we have:

One determines \(\varphi_{i,0,0}\) step by step, taking into account that \(\varphi_{0,0,0}\) is given by formula (5.2.13). One determines \(\varphi_{1,0,0}\), then \(\varphi_{2,0,0}, \ldots\), and so on, as in subsections 4.2.4 and 4.2.5. One finds

for \(r = 1, 2, \ldots, n\).

Next, suppose that we have obtained the formula

valid for \(r = 1, 2, \ldots, n\), which is correct for \(i = 1\), since in this case formula (5.2.19) coincides with formula (5.2.18).

To prove that formula (5.2.19) is valid for \(i = 1, 2, \ldots, n\), we use the method of complete induction. To this end one integrates the partial differential equation (5.2.14) corresponding to the index \(i + 1\), with the boundary conditions (5.2.15), (5.2.16), (5.2.17) corresponding to the index \(i + 1\), taking formula (5.2.19) into account. In this way one obtains a formula for \(\dfrac{\partial^{3(n-r)} \varphi_{i+1,0,0}}{\partial x^{n-r} \partial y^{n-r} \partial z^{n-r}}\), valid for \(r = 1, 2, \ldots, n\), which is deduced from formula (5.2.19) by replacing \(i\) by \(i + 1\). This proves that formula (5.2.19) is general; it is valid for \(i = 1, 2, \ldots, n-1\).

For \(i = n - 1\) and \(r = n\), formula (5.2.19) gives us

It remains to determine the constants \(C_{0,0,0}, C_{1,0,0}, \ldots, C_{n,0,0}\) by conditions (5.2.15), (5.2.16), (5.2.17). For \(i = n\), and taking convention (A) into account, the conditions reduce to:

for \(r = 0, 1, \ldots, n-1\).

For \(r = 0\), formula (5.2.20) shows that the first three conditions (5.2.21) are satisfied, and the fourth condition gives the equation

Formula (5.2.20) shows that

and in order that the first condition (5.2.21) be satisfied when \(r = 1, 2, \ldots, n-1\), we must take

for \(r = 1, 2, \ldots, n-1\).

On the other hand, we have

and it follows from equations (5.2.23) that the other conditions (5.2.21) are satisfied for \(r = 1, 2, \ldots, n-1\).

Hence the constants \(C_{0,0,0}, C_{1,0,0}, \ldots, C_{n,0,0}\) satisfy equations (5.2.22) and (5.2.23), which can also be written in the form

One shows as in subsection 4.2.16 that \(\varphi_{n-1,0,0}\) can also be written in the form

5.2.4. If in conditions (5.2.5) and (5.2.6) we set \(i = 0\), \(k = 0\) and take convention (A) into account, we have

Proceeding as in subsection 5.2.3, one proves the formula

valid for \(j = 1, 2, \ldots, n-1\) and \(r = 1, 2, \ldots, n\).

One shows that the constants \(C_{0,0,0}, C_{0,1,0}, \ldots, C_{0,n,0}\) are determined by the equations

and

for \(r = 1, 2, \ldots, n-1\). This system can be written in the form

These equations show that, for \(r = n\) and \(j = n-1\), formula (5.2.30) can be written in the form

5.2.5. If in conditions (5.2.5) and (5.2.6) we set \(i = 0\), \(j = 0\) and take convention (A) into account, we obtain:

Proceeding as in subsection 5.2.3, one proves the formula

valid for \(r = 1, 2, \ldots, n\) and \(k = 1, 2, \ldots, n-1\).

The constants \(C_{0,0,0}, \ldots, C_{0,0,n}\) are given by the equations

for \(r = 1, 2, \ldots, n-1\). This system can also be written in the form

Equations (5.2.41) show that, for \(r = n\) and \(k = n-1\), formula (5.2.39) can be written in the form

§ 5.3. The problems \((P_{i,j,0})\), \((P_{0,j,k})\), \((P_{i,0,k})\)

5.3.1. If in the boundary conditions (5.2.5) and (5.2.6) we set \(k = 0\) and take convention (A) into account, we obtain:

5.3.2. We first determine the function \(\varphi_{1,1,0}\) from the partial differential equation (5.3.1) and the boundary conditions (5.3.2), (5.3.3), (5.3.4) corresponding to \(i = 1\), \(j = 1\). For this we must take into account formula (5.2.13), formula (5.2.19) for \(i = 1\) and formula (5.2.30) for \(j = 1\). Then, proceeding as in subsection 4.2.8, we find

for \(r = 1, 2, \ldots, n\).

Next, by the method of complete induction, as in subsection 4.2.9, one proves the formula

for \(i = 1, 2, \ldots n-1\) and \(r = 1, 2, \ldots n\).

It remains to determine the constants \(C_{0,1,0}, \ldots, C_{n,1,0}\) from conditions (5.3.2), (5.3.3), (5.3.4) corresponding to the indices \(i = n\) and \(j = 1\). Taking convention (A) into account, these conditions are:

By formulas (5.3.6) and (5.2.20) we have

and

For \(r = 0\), conditions (5.3.7) are:

The first three conditions (5.3.10) are satisfied, since \(\varphi_{n-1,0,0}\) is a polynomial of degree \(n-1\) in \(x\), in \(y\) and in \(z\).

The last condition (5.3.10) leads to the equation

By formula (5.3.8) we have

But formula (5.2.23) shows that the first large bracket on the right-hand side vanishes for \(r = 1, 2, \ldots, n-1\). Then, for the first condition to be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

Finally, we have

and from formulas (5.2.23) and (5.3.12) one sees that for \(r = 1, 2, \ldots, n-1\) the right-hand sides of these formulas vanish. Hence the remaining conditions (5.3.7) are satisfied for \(r = 1, 2, \ldots, n-1\).

Thus all the conditions (5.3.7) are satisfied, and the constants \(C_{0,1,0}, \ldots, C_{n,1,0}\) are determined by equations (5.3.11) and (5.3.12). The system formed by these equations can also be written in the form

Let us suppose that we have proved that for \(l = 1, 2, \ldots, j\) we have the formula

for \(l = 0, 1, 2, \ldots, j\), \(i = 1, 2, \ldots, n-1\) and \(r = 1, 2, \ldots, n\), where the constants \(C_{0,l,0} \ldots, C_{n,l,0}\) are determined by the equations

for \(r = 1, 2, \ldots, n-1\). This system can also be written in the form

Systems (5.3.16), (5.3.17) are valid for \(l = 0, 1, \ldots, j\).

This is true for \(l = 1\), because in this case formula (5.3.15) coincides with formula (5.3.6) and system (5.3.17) coincides with system (5.3.14).

We shall prove that formula (5.3.15) also holds for \(i = 1, 2, \ldots, n-1\), \(l = j + 1\) and \(r = 1, 2, \ldots, n\), the constants \(C_{0,j+1,0}, \ldots, C_{n,j+1,0}\) being given by system (5.3.16) or (5.3.17), for \(l = j + 1\).

To this end we integrate the partial differential equations

with the boundary conditions (5.3.2), (5.3.3), (5.3.4) corresponding to the indices \(i\), \(j + 1\).

We begin by determining \(\frac{\partial^{3(n-r)}\varphi_{1,j+1,0}}{\partial x^{n-r}\partial y^{n-r}\partial z^{n-r}}\) for \(r = 1, 2, \ldots, n\), integrating the equation \(\frac{\partial^{3n}\varphi_{1,j+1,0}}{\partial x^n \partial y^n \partial z^n} = 0\) with the boundary conditions specified above for the indices \(1, j + 1\), and taking into account formula (5.3.15) for \(i = 1\), \(l = j\) and formula (5.2.30), which gives \(\frac{\partial^{3(n-r)}\varphi_{0,j,0}}{\partial x^{n-r}\partial y^{n-r}\partial z^{n-r}}\) for \(j = 1, 2, \ldots, n-1\) and \(r = 1, 2, \ldots, n\). Applying the method of complete induction, one finds the formula

next, one assumes that we have the formula

valid for \(i' = 1, 2, \ldots, i\), which is true for \(i' = 1\), because in this case formula (5.3.20) coincides with formula (5.3.19). We shall prove that formula (5.3.20) also holds for \(i' = i + 1\). To this end we shall integrate the partial differential equation

with the boundary conditions (5.3.2), (5.3.3), (5.3.4) corresponding to the indices \(i + 1\), \(j + 1\), \(0\).

To write these conditions one uses formula (5.3.20) for \(i' = i\) and formula (5.3.15) for \(l = j\). One arrives at the formula giving the derivative \(\frac{\partial^{3(n-r)}\varphi_{i+1,j+1,0}}{\partial x^{n-r}\partial y^{n-r}\partial z^{n-r}}\), valid for \(r = 1, 2, \ldots, n\). One finds that this formula is obtained from formula (5.3.20) by replacing \(i'\) by \(i + 1\).

It follows that formula (5.3.20) is general; it is valid for \(i' = 1, 2, \ldots, n-1\). But formula (5.3.20) is obtained from formula (5.3.15) by replacing \(l\) by \(j+1\). Formula (5.3.15) is therefore general; it is valid for \(i = 1, 2, \ldots, n-1\) and \(l = 1, 2, \ldots, n-1\).

It remains to show how the constants \(C_{0,j+1,0}, \ldots, C_{n,j+1,0}\) are determined.

5.3.4. Taking convention (A) into account, the boundary conditions (5.3.2), (5.3.3), (5.3.4) for \(i = n\) are

Setting \(i = n-1\), \(l = j+1\), \(r = n\) in formula (5.3.15), we have

and further, replacing \(j + 1\) by \(j\), we have

For \(r = 0\), the first three conditions (5.3.21) are satisfied, because \(\varphi_{n-1,j+1,0}\) and \(\varphi_{n-1,j,0}\) are polynomials of degree \(n-1\) in \(z\) and \(y\).

The fourth condition gives the equation

We have

and since equation (5.3.16) is satisfied for \(r = 1, 2, \ldots, n-1\) and \(l = 1, 2, \ldots, j\), the preceding formula reduces to

For the first condition (5.3.21) to be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

which shows that the constants \(C_{0,j+1,0}, \ldots, C_{n-1,j+1,0}\) satisfy system (5.3.16), i.e. system (5.3.17).

It remains to see that the other conditions (5.3.21) are also satisfied.

We have

and the right-hand side vanishes because, by equations (5.3.16), each large bracket vanishes for \(r = 1, 2, \ldots, n-1\) and \(l = 0, 1, \ldots, j, j+1\). Hence conditions (5.3.21) are satisfied for \(r = 1, 2, \ldots, n-1\).

We have

and the right-hand side vanishes, as equations (5.3.25) show, for \(r = 1, 2, \ldots, n-1\). Hence conditions (5.3.21) are satisfied for \(r = 1, 2, \ldots, n-1\).

Finally, we have

and therefore the last condition (5.3.21) is satisfied for \(r = 1, 2, \ldots, n-1\).

5.3.5. If in formula (5.3.15) we set \(i = n-1\) and take systems (5.3.17) into account, for \(l = 0, 1, \ldots, j\) we obtain the formula

which will be used later; it is valid for \(j = 0, 1, \ldots, n-1\).

5.3.6. We wish to show that the constants \(C_{i,j,0}\) also satisfy the systems of equations

for \(r = 1, 2, \ldots, n-1\) and \(i = 0, 1, \ldots, n-1\), i.e. the system of equations

for \(i = 0, 1, \ldots, n-1\).

The system formed by equations (5.3.30) and (5.3.31) is true for \(i = 0\), because in this case it coincides with the system formed by equations (5.2.31) and (5.2.32).

To this end we set \(j = n\) in the boundary conditions (5.3.2), (5.3.3), (5.3.4) and take convention (A) into account. We obtain the conditions

for \(r = 0, 1, \ldots, n-1\).

We write formulas (5.3.15) for \(l = n-1\) in the form

valid for \(i = 0, 1, \ldots, n-1\) and \(r = 1, 2, \ldots, n\).

Setting \(r = 0\) in formulas (5.3.33), we obtain the conditions

The first three conditions are satisfied because \(\varphi_{i,n-1,0}\) and \(\varphi_{i-1,n-1,0}\) are polynomials of degree \(n-1\) in \(x\), in \(y\) and in \(z\).

The last condition (5.3.35) gives the equation

valid for \(i = 0, 1, \ldots, n-1\).

Let us now examine conditions (5.3.33) for \(r = 1, 2, \ldots n-1\). We have

and therefore

The first large bracket on the right-hand side vanishes for \(r = 1, 2, \ldots, n-1\) by equations (5.2.32), and for the first conditions (5.3.33) to be satisfied for \(r = 1, 2, \ldots, n-1\) and \(i = 1, 2, \ldots, n-1\), equations (5.3.31) must hold for \(r = 1, 2, \ldots, n-1\). Likewise, we have

and one sees from equations (5.3.31) that the other conditions (5.3.33) are satisfied for \(r = 1, 2, \ldots, n-1\).

5.3.7. Let us now prove that the system of equations

which is valid for \(j = 0, 1, \ldots, n-1\), is also valid for \(j = n\), and that the system of equations

which is valid for \(i = 0, 1, \ldots, n-1\), is also valid for \(i = n\).

To this end we observe that the formula

can be written, by equations (5.3.17), for \(i = n-1\), in the form

which is valid for \(j = 0, 1, \ldots, n-1\).

Likewise, formula (5.3.36) can be written, by equations (5.3.32), for \(j = n-1\), in the form

Finally, we observe that, setting \(i = n-1\) in formula (5.3.38) and \(j = n-1\) in formula (5.3.37), the function \(\varphi_{n-1,n-1,0}\) can be written in the form

or in the form

Let us write the boundary conditions (5.3.2), (5.3.3), (5.3.4) for \(i = n\) and \(j = n\). By convention (A), these conditions are

and

For \(r = 0\), condition (5.3.41) is satisfied, and condition (5.3.42) leads, by formula (5.3.39), to the equation

and, by formula (5.3.40), to the equation

Let us now consider conditions (5.3.2), (5.3.3), (5.3.4) for \(i = n\), \(j = n-1\) and then for \(i = n-1\), \(j = n\). Of these conditions we take only the following:

and

which are satisfied for \(r = 0\).

Using formula (5.3.39) we have

and, using formula (5.3.40), we have

For conditions (5.3.45), (5.3.46) to be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

and

The system formed by equations (5.3.43) and (5.3.44) shows that system (5.3.17) is also valid for \(j = n\).

The system formed by equations (5.3.44) and (5.3.48) shows that system (5.3.32) is also valid for \(i = n\).

Let us observe, finally, that by formula (5.3.39) we have

and

and by equations (5.3.47) it follows that conditions (5.3.41) and (5.3.42) are satisfied for \(r = 1, 2, \ldots, n-1\).

5.3.8. In summary, the solution of the boundary value problem posed in subsection 5.3.1 is as in formula (5.3.15), i.e.

where the constants \(C_{i,j,0} (i, j = 0, 1, \ldots, n)\) satisfy the systems of equations

for \(j = 0, 1, \ldots, n\) and

for \(i = 0, 1, \ldots, n\).

Taking these systems into account, formulas (5.3.39) and (5.3.40) can be written in the form

5.3.9. If in the boundary conditions (5.2.5) and (5.2.6) we set \(i = 0\) and take convention (A) into account, we obtain

Proceeding as in subsections 5.3.2 – 5.3.8, one proves that the solution of this problem is given by the formula

where the constants \(C_{0,j,k}\) satisfy, for \(j, k = 0, 1, \ldots, n\), the systems of equations

and

Taking these systems into account, one can write

5.3.10. If in the boundary conditions (5.2.5) and (5.2.6) we set \(j = 0\) and take convention (A) into account, we obtain

Proceeding as in subsections 5.3.2 – 5.3.8, one proves that the solution of this problem is given by the formula

where the constants \(C_{i,0,k}\) satisfy, for \(i, k = 0, 1, \ldots, n\), the systems of equations

and

§ 5.4. The problem \((P_{i,j,1})\)

In this section we determine the solution \(\varphi_{i,j,1}\) of problem \((P_{i,j,1})\), following the same method as the one used to determine the function \(\varphi_{i,j,0}\), but with new boundary conditions which must be satisfied by the functions \(\varphi_{i,j,1}\). The functions \(\varphi_{i,j,0}\), \(\varphi_{i,0,k}\), \(\varphi_{0,j,k}\), \(\varphi_{i,0,0}\), \(\varphi_{0,j,0}\), \(\varphi_{0,0,k}\), \(\varphi_{0,0,0}\) determined in the preceding sections enter here, but with new boundary conditions for \(\varphi_{i,j,1}\). This section is an introduction to the general case.

5.4.1. Let us first determine the function \(\varphi_{1,1,1}\)

For this we must integrate the partial differential equation

with the boundary conditions (5.2.5), (5.2.6), (5.2.7) corresponding to the indices \(i = 1\), \(j = 1\), \(k = 1\). These conditions are

further,

Finally,

We can solve this problem because in § 5.2 and § 5.3 we have determined the functions \(\varphi_{0,0,0}\), \(\varphi_{1,0,0}\), \(\varphi_{0,1,0}\), \(\varphi_{0,0,1}\), \(\varphi_{1,1,0}\), \(\varphi_{1,0,1}\), \(\varphi_{0,1,1}\).

We found

Then, taking these formulas into account, it follows that

Integrating equation (5.4.1) with these conditions, one finds

Next, we must integrate equation (5.4.1) with the boundary conditions (5.4.2), (5.4.3), (5.4.4). For this it is useful to write down the following formulas, which follow from formulas (5.4.5), (5.4.6), (5.4.7):

Then, by the boundary conditions (5.4.2), (5.4.3), (5.4.4), for \(r = 1\) we have

Integrating the partial differential equation (5.4.8) with these conditions, one finds

Let us suppose we have proved that for the index \(r\) we have

which is true for \(r = 2\), because in this case formula (5.4.11) coincides with formula (5.4.10).

We shall prove that formula (5.4.11) also holds for the index \(r + 1\).

To this end we shall integrate equation (5.4.11) with the boundary conditions (5.4.2), (5.4.3), (5.4.4) relative to the index \(r\). By formulas (5.4.9) we shall have

Integrating the partial differential equation (5.4.10) with these conditions, one finds

Formula (5.4.12) is obtained from formula (5.4.11) by replacing \(r\) by \(r + 1\), which shows that formula (5.4.11) is general; it is valid for \(r = 1, 2, \ldots, n-1\).

We have presented the computation of \(\varphi_{1,1,1}\) in detail in order to show the method that will be used routinely in this work.

5.4.2. We shall now prove the formula

valid for \(r = 1, 2, \ldots, n-1\) and \(i = 1, 2, \ldots, n-1\).

We use the method of complete induction and observe first that formula (5.4.13) is valid for \(i = 1\), since in this case it coincides with formula (5.4.10). To prove it in general, we shall assume that formula (5.4.13) is true for the index \(i\) and prove that it is also valid for the index \(i + 1\).

To this end we shall integrate the partial differential equation

with the boundary conditions (5.4.2), (5.4.4) corresponding to the indices \(i + 1, 1, 1\) and for \(r = 0, 1, \ldots, n-1\).

For \(r = 0\), we are led to the equation

Next, integrating equation (5.4.15) with conditions (5.4.2), (5.4.3), (5.4.4) for \(r = 1\), one finds

By the method of complete induction one proves the formula

This formula is true for \(r = 2\), because in this case it coincides with formula (5.4.16). Assuming that formula (5.4.17) is true for the index \(r\), one proves that it is also true for the index \(r + 1\) by integrating equation (5.4.17) with conditions (5.4.2), (5.4.3), (5.4.4) corresponding to the index \(r\). One arrives at a formula which can be obtained from formula (5.4.17) by replacing \(r\) by \(r + 1\). This proves that formula (5.4.13) is general; it is valid for \(i = 0, 1, \ldots, n-1\) and \(r = 1, 2, \ldots, n-1\).

5.4.3. To determine the constants \(C_{0,1,1} \ldots, C_{n,1,1}\) we set \(i = n\) in conditions (5.2.5), (5.2.6), (5.2.7). We obtain the conditions

Setting \(i = n-1\) in formulas (5.4.13), (5.3.15), (5.3.62), (5.2.20), we obtain

To these equations we add the following systems of equations:

for \(r = 1, 2, \ldots, n-1\), which follow from equations (5.3.12), (5.3.63) and (5.2.23).

For \(r = 0\), the first three conditions (5.4.18) are satisfied because \(\varphi_{n-1,1,1}\), \(\varphi_{n-1,1,0}\), \(\varphi_{n-1,0,1}\), \(\varphi_{n-1,0,0}\) are polynomials of degree \(n-1\) in \(y\) and in \(z\).

The last condition (5.4.18) leads to the equation

By the first equation (5.4.19) we have

But by equations (5.4.20), valid for \(r = 1, 2, \ldots, n-1\), the first three large brackets of the preceding formula vanish, and there remains:

This formula shows that the first condition (5.4.18) is satisfied for \(r = 1, 2, \ldots, n-1\) if we take

We have the formulas

These formulas show that, by systems (5.4.20) and (5.4.22), valid for \(r = 1, 2, \ldots, n-1\), the other conditions (5.4.18) are satisfied for \(r = 1, 2, \ldots, n-1\).

We have thus proved that the constants \(C_{0,1,1}, \ldots, C_{n,1,1}\) are determined by equations (5.4.21) and (5.4.22). These constants satisfy the system of equations

5.4.4. Let us suppose that we have proved the formula

valid for \(r = 1, 2, \ldots, n\); \(i = 0, 1, \ldots, n-1\) and \(l = 1, 2, \ldots, j\).

In this formula there appear the constants \(C_{0,l,0}, \ldots, C_{n,l,0}\), which satisfy the systems of equations

for \(r = 1, 2, \ldots, n-1\) and \(l = 0, 1, \ldots, n\); this system reduces to the system

(See subsections 5.3.3, 5.3.4, 5.3.7, 5.3.8.)

We assume that the constants \(C_{0,l,1}, \ldots, C_{n,l,1}\) satisfy the system of equations

for \(r = 1, 2, \ldots, n-1\) and \(l = 0, 1, \ldots, j\). This system reduces to the system

For \(l = 0\), system (5.4.27) reduces to the system formed by equations (5.3.11) and (5.3.12). For \(l = 1\), system (5.4.27) reduces to the system formed by equations (5.4.21) and (5.4.22).

We wish to prove that formula (5.4.24) still holds for \(l = j + 1\) and that the constants \(C_{0,j+1,1}, \ldots, C_{n,j+1,1}\) satisfy system (5.4.27), or (5.4.28), for \(l = j + 1\). For this we must integrate the partial differential equation

with the boundary conditions (5.2.5), (5.2.6) and (5.2.7) for \(r = 0, 1, \ldots, n-1\), which correspond to the indices \(i, j + 1, 1\).

5.4.5. Let us begin by determining \(\varphi_{1,j+1,1}\). For this we must integrate the partial differential equation (5.4.29) with the boundary conditions (5.2.5), (5.2.6), (5.2.7) for \(r = 0, 1, \ldots, n-1\), which correspond to the indices \(1, j + 1, 1\).

Proceeding as in subsection 5.4.2 to prove formula (5.4.13), we obtain the formula

which can also be written in the form

and which is valid for \(r = 1, 2, \ldots, n-1\) and \(j = 0, 1, \ldots, n-2\).

5.4.6. Let us suppose that we have proved the formula

valid for \(r = 1, 2, \ldots, n\) and \(h = 0, 1, \ldots, i\). This is true for \(i = 0\) and \(i = 1\), because in these cases formula (5.4.32) coincides with formula (5.3.54) (and (5.4.31)).

Let us prove that formula (5.4.32) also holds for \(h = i + 1\).

For this we must integrate the partial differential equation

with the boundary conditions (5.2.5), (5.2.6) and (5.2.7) corresponding to the indices \(i + 1, j + 1, 1\) and for \(r = 0, 1, \ldots, n-1\).

For \(r = 0\), one finds

For \(r = 1\), one finds

Let us prove that in general we have

Formula (5.4.36) is true for \(r = 1\) and \(r = 2\), because in these cases it coincides with formulas (5.4.34), (5.4.35). To prove that formula (5.4.36) is true in general, we assume that it is true for the index \(r\) and prove that it is also true for the index \(r + 1\). For this we integrate equation (5.4.36) with the boundary conditions (5.2.5), (5.2.6) and (5.2.7) corresponding to the indices \(i + 1, j + 1, 1\) and to the index \(r\).

One finds

Formula (5.4.37) is obtained from formula (5.4.36) by replacing \(r\) by \(r + 1\). It follows that formula (5.4.36) is general; it is valid for \(r = 1, 2, \ldots, n-1\).

On the other hand, formula (5.4.36) is obtained from formula (5.4.32) by replacing \(h\) by \(i + 1\). It follows that formula (5.4.32) is general; it is valid for \(h = 0, 1, \ldots, n-1\).

5.4.6. We have thus proved the formula

valid for \(r = 1, 2, \ldots, n\) and \(i, j = 0, 1, \ldots, n-1\).

It remains to prove that the constants \(C_{0,j+1,1}, \ldots, C_{n,j+1,1}\) satisfy system (5.4.27) or (5.4.28).

For this we write that conditions (5.2.5), (5.2.6) and (5.2.7) are satisfied when we set \(i = n\), \(k = 1\) and take convention (A) into account.

We shall have the following conditions:

for \(r = 0, 1, \ldots, n-1\).

For \(r = 0\), the first three conditions (5.4.39) are satisfied because \(\varphi_{n-1,j+1,1}\), \(\varphi_{n-1,j,0}\), \(\varphi_{n-1,j,1}\) are polynomials of degree \(n-1\) in \(y\) and in \(z\).

The fourth condition (5.4.39) leads to the equation

We have the formula

But by equations (5.4.25) and (5.4.27) all the large brackets on the right-hand side, except the last, vanish for \(r = 1, 2, \ldots, n-1\). Then, for the first condition (5.4.39) to be satisfied, we must take

for \(r = 1, 2, \ldots, n-1\), which, together with equation (5.4.40), proves that equations (5.4.27) also hold for \(l = j + 1\).

It remains for us to prove that the last three conditions (5.4.39) are satisfied for \(r = 1, 2, \ldots, n-1\).

By formulas (5.4.38) and (5.3.36) we have

These formulas show that, by equations (5.4.25) and (5.4.42), all the large brackets on the right-hand side vanish for \(r = 1, 2, \ldots, n-1\), which proves that the other boundary conditions (5.4.39) are satisfied for \(r = 1, 2, \ldots, n-1\).

5.4.7. We have proved that the constants \(C_{i,j,0}\) and \(C_{i,j,1}\) satisfy the systems of equations (5.4.26) and (5.4.28):

for \(j = 0, 1, \ldots, n\), and

for \(j = 0, 1, \ldots, n-1\).

We have also proved that the constants \(C_{i,j,0}\) satisfy the systems of equations

for \(r = 1, 2, \ldots, n-1\) and \(i = 0, 1, \ldots, n\), i.e. the system of equations

for \(i = 0, 1, \ldots, n\).

We wish to prove that the constants \(C_{i,j,1}\) satisfy the system of equations

for \(r = 1, 2, \ldots, n-1\) and \(i = 0, 1, \ldots, n-1\), and

for \(i = 0, 1, \ldots, n-1\).

Let us set \(j = n\), \(k = 1\) in the boundary conditions (5.2.5), (5.2.6), (5.2.7) and take convention (A) into account. These conditions will be

for \(r = 0, 1, \ldots, n-1\).

Formula (5.4.38), for \(j = n-1\) and \(r = n\), can be written in the form

and we have analogous formulas for \(\varphi_{i-1,n-1,1}\), \(\varphi_{i,n-1,0}\) and \(\varphi_{i-1,n-1,0}\), which we shall not write down.

Let us consider conditions (5.4.47) for \(r = 0\). The first three are satisfied, because the functions \(\varphi_{i,n-1,1}\), \(\varphi_{i-1,n-1,1}\), \(\varphi_{i,n-1,0}\), \(\varphi_{i-1,n-1,0}\) are polynomials of degree \(n-1\) in \(x\) and \(z\).

The fourth condition (5.4.47) leads to the equation

Let us now examine conditions (5.4.47) for \(r = 1, 2, \ldots, n-1\).

We have

But by equations (5.4.45) the first \(i+1\) large brackets vanish for \(r = 1, 2, \ldots, n-1\).

On the other hand, for the first condition (5.4.47) to be satisfied for \(i = 0\), we must take in formula (5.4.50)

which, together with equation (5.4.49) for \(i = 0\), leads to system (5.3.55) for \(k = 1\), which we have already considered.

Further, for the first condition (5.4.47) to be satisfied for \(i = 1\), we must take in formula (5.4.50)

Step by step one proves that we have the equations

and \(i = 0, 1, \ldots, n-1\).

We have thus proved that the constants \(C_{i,j,1}\) satisfy the systems of equations (5.4.45) for \(r = 1, 2, \ldots, n-1\) and \(i = 0, 1, \ldots, n-1\), i.e. the systems of equations (5.4.46) for \(i = 0, 1, \ldots, n-1\).

It remains to show that the other boundary conditions (5.4.47) are satisfied for \(r = 1, 2, \ldots, n-1\).

For this we observe that we can write

Equations (5.3.31) and (5.4.45) show that the other equations (5.4.47) are satisfied for \(r = 1, 2, \ldots, n-1\).

5.4.8. Formula (5.4.31), in which we set \(i = n-1\), becomes

But by systems (5.4.43) and (5.4.44) we can also write formula (5.4.51) in the form

Later on we shall need this formula for \(j = n-1\), i.e.

In the same way, if in formula (5.4.38) we set \(j = n-1\) and use systems (5.3.32) and (5.4.46), we have

and later on we shall need this formula for \(i = n-1\), i.e.

5.4.9. To prove that the systems of equations (5.4.44) and (5.4.46) also hold for \(j = n\) and \(i = n\), let us write that conditions (5.2.5), (5.2.6), (5.2.7) are satisfied when we set \(i = n\), \(j = n\), \(k = 1\), taking convention (A) into account.

We shall have the following conditions:

To these conditions we add conditions (5.2.5), (5.2.6), (5.2.7) for \(i = n\), \(j = n-1\), \(k = 1\) and then for \(i = n-1\), \(j = n\), \(k = 1\). Of these equations we take only the following:

We shall use formulas (5.4.53), (5.4.55) as well as the formulas

If we consider conditions (5.4.56) for \(r = 0\), the first is satisfied because \(\varphi_{n-1,n-1,1}\) is a polynomial of degree \(n-1\) in \(z\). If we take formulas (5.4.55), (5.3.39) into account, the second condition (5.4.56) leads to the equation

and if we take formulas (5.4.53), (5.3.40) into account, the second condition (5.4.56) leads to the equation

Conditions (5.4.57), (5.4.58) are satisfied for \(r = 0\), because \(\varphi_{n-1,n-1,1}\) is a polynomial of degree \(n-1\) in \(x\), in \(y\) and in \(z\).

It remains to examine these conditions for \(r = 1, 2, \ldots, n-1\).

According to formulas (5.4.55) and (5.4.53) we have

and

But we have proved that

and

for \(r = 1, 2, \ldots, n-1\), so that formulas (5.4.61) and (5.4.62) reduce to

and

For conditions (5.4.57) and (5.4.58) to be satisfied for \(r = 1, 2, \ldots, n-1\), we must take

and

Equations (5.4.59) and (5.4.63) for \(r = 1, 2, \ldots, n-1\) show that the system of equations (5.4.44) also holds for \(j = n\).

Likewise, equations (5.4.60) and (5.4.64) for \(r = 1, 2, \ldots, n-1\) show that the system of equations (5.4.46) also holds for \(i = n\).

It remains to prove that conditions (5.4.56) are satisfied for \(r = 1, 2, \ldots, n-1\).

Indeed, by equations (5.4.55) we have

and

But equations (5.3.47) and (5.4.63) show that the large brackets on the right-hand sides of formulas (5.4.65), (5.4.66) vanish for \(r = 1, 2, \ldots, n-1\), which proves that conditions (5.4.56) are satisfied for \(r = 1, 2, \ldots, n-1\).

5.4.10. In summary, the solution of the boundary value problem \((P_{i,j,1})\) is given, by formulas (5.4.38), by

for \(i, j = 0, 1, \ldots, n-1\).

The constants \(C_{i,j,1}\) satisfy the systems of equations

for \(j = 0, 1, \ldots, n\) and

for \(i = 0, 1, \ldots, n\).

§ 5.5. The general case: the problem \((P_{i,j,k})\)

5.5.1. Let us suppose that we have proved the formula

for \(i, j = 0, 1, \ldots, n-1\) and \(l = 0, 1, \ldots, k\).

Formula (5.5.1) is true for \(l = 0\) and for \(l = 1\), because in these cases it coincides with formulas (5.3.15) and (5.4.67).

We wish to prove that formula (5.5.1) also holds for \(l = k+1\), and therefore that it holds for \(l = 0, 1, \ldots, n-1\).

We also assume that the constants \(C_{i,j,l}\) satisfy the systems of equations

for \(r = 1, 2, \ldots, n-1\), \(j = 0, 1, \ldots, n-1\) and \(l = 0, 1, \ldots, k\), i.e. the systems

for \(j = 0, 1, \ldots, n-1\) and \(l = 0, 1, \ldots, k\).

We assume, moreover, that the constants \(C_{i,j,l}\) satisfy the systems of equations

for \(r = 1, 2, \ldots, n-1\), \(i = 0, 1, \ldots, n\) and \(l = 0, 1, \ldots, k\), i.e. the systems

for \(i = 0, 1, \ldots, n-1\) and \(l = 0, 1, \ldots, k\).

This is true for \(l = 0\), because in this case systems (5.5.3), (5.5.5) coincide with systems (5.3.17), (5.3.32).

It is also true for \(l = 1\), because in this case systems (5.5.3), (5.5.5) coincide with systems (5.4.68), (5.4.69).

We wish to prove that systems (5.5.2), (5.5.4), i.e. (5.5.3), (5.5.5), also hold for \(l = k+1\), and therefore that they hold for \(l = 0, 1, \ldots, n-1\).

We shall further prove that systems (5.5.3), (5.5.5) also hold for \(l = n\).

5.5.2. To solve the problems posed in subsection 5.5.1 we use the method of complete induction.

We shall integrate the partial differential equation

with the boundary conditions (5.2.5), (5.2.6), (5.2.7) corresponding to the indices \(i\), \(j\), \(k+1\), i.e.

for \(i, j = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Further,

for \(i, j = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

Finally,

for \(i, j = 0, 1, \ldots, n\) and \(r = 0, 1, \ldots, n-1\).

5.5.3. Let us begin by determining \(\varphi_{1,1,k+1}\). For this we shall integrate the partial differential equation

with the boundary conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(i = 1\), \(j = 1\).

These conditions involve the functions

The first of these functions is given by formula (5.5.1) for \(i = 1\), \(j = 1\) and \(l = k\). The functions \(\varphi_{0,0,k}\) and \(\varphi_{0,0,k+1}\) are the solutions of problems \((P_{0,0,k})\), \((P_{0,0,k+1})\) treated in subsection 5.2.5. The functions \(\varphi_{1,0,k}\) and \(\varphi_{1,0,k+1}\) are the solutions of problems \((P_{1,0,k})\), \((P_{1,0,k+1})\) treated in subsection 5.3.10. The functions \(\varphi_{0,1,k}\) and \(\varphi_{0,1,k+1}\) are the solutions of problems \((P_{0,1,k})\), \((P_{0,1,k+1})\) treated in subsection 5.3.9. All these functions can be deduced from formula (5.5.1).

Integrating the partial differential equation (5.5.10) with conditions (5.5.7), (5.5.8), (5.5.9) corresponding to \(r = 0\), one finds

Next, integrating equation (5.5.11) with conditions (5.5.7), (5.5.8), (5.5.9) corresponding to \(r = 1\), one finds

Let us suppose that we have proved the formula

This formula is true for \(r = 1\) and \(r = 2\), because in these cases it coincides with formulas (5.5.11) and (5.5.12). Formula (5.5.13) is also true for the index \(r + 1\), because, integrating equation (5.5.13) with conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(1, 1, k + 1\) and to the index \(r\), one obtains formula (5.5.13) with the index \(r\) replaced by \(r + 1\).

Formula (5.5.13) is therefore general; it is valid for \(r = 1, 2, \ldots, n\).

5.5.4. We shall now prove the formula

for \(r = 1, 2, \ldots, n\) and \(i' = 0, 1, \ldots, n-1\).

We observe first that formula (5.5.14) is true for \(i' = 1\), because in this case it coincides with formula (5.5.13). Next we assume that formula (5.5.14) is true for the index \(i' = i\), and we shall prove that it is also true for the index \(i' = i + 1\).

To carry out the proof we shall integrate the partial differential equation

with the boundary conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(i + 1, 1, k + 1\) and for \(r = 0, 1, \ldots, n-1\).

These conditions involve the functions

The functions \(\varphi_{i+1,1,k}\), \(\varphi_{i,1,k}\) are given by formula (5.5.1) for the indices \(i + 1, 1, k\) and \(i, 1, k\). The functions \(\varphi_{i,0,k}\), \(\varphi_{i,0,k+1}\), \(\varphi_{i+1,0,k}\), \(\varphi_{i+1,0,k+1}\) are the solutions of problems \((P_{i,0,k})\), \((P_{i,0,k+1})\), \((P_{i+1,0,k})\), \((P_{i+1,0,k+1})\), which we treated in subsection 5.3.10. Finally, the function \(\varphi_{i,1,k+1}\) is given by formula (5.5.14) for \(r = n\), which we have assumed valid for \(i' = i\).

Integrating equation (5.5.15) with the boundary conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(i + 1, 1, k + 1\) and for \(r = 0\), one finds

Next, integrating equation (5.5.16) with the boundary conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(i + 1, 1, k + 1\) and for \(r = 1\), one finds

Let us suppose that we have proved the formula

This formula is true for \(r = 2\), because in this case it coincides with formula (5.5.17). Formula (5.5.18) is also valid for the index \(r + 1\), because, integrating equation (5.5.18) with the boundary conditions (5.5.7), (5.5.8), (5.5.9) corresponding to the indices \(i + 1, 1\), \(k + 1\) and to the index \(r\), one obtains formula (5.5.18) with \(r\) replaced by \(r + 1\).

Formula (5.5.18) is therefore general; it is valid for \(r = 1, 2, \ldots, n\).

On the other hand, formula (5.5.18) is obtained from formula (5.5.14) by replacing \(i'\) by \(i + 1\), which means that formula (5.5.14) is general; it is valid for \(i' = 0, 1, \ldots, n-1\).

It remains to determine the constants \(C_{0,1,k+1}, \ldots, C_{n,1,k+1}\).

5.5.5. Let us write conditions (5.5.7), (5.5.8), (5.5.9) for the indices \(n, 1, k + 1\), taking convention (A) into account. These conditions are:

for \(r = 0, 1, \ldots, n-1\).

By formula (5.5.14), for \(i' = n - 1\) we have

and by formula (5.3.62) we have