ABOUT THE REST IN SOME LINEAR APPROXIMATION FORMULAS OF ANALYSIS

by
TIBERIU POPOVICIU
(Cluj)

Many of the approximation formulas of analysis are of the form

A[f]\approx B[f],\text{ sau }A[f]=B[f]+R[f],

(*)

where $A[f],B[f]$ are linear functionals defined on a vector set of real and continuous functions of a real variable and whose remainder $R[f]=A[f]--B[f]$ is canceled on $n+1$ given functions $f_{i},i=0,1,\ldots,n$ The rest $R[f]$ is also a linear functional and vanishes on any linear combination of the functions $f_{i}$ .

We will only consider real functionals and by a linear functional we will understand an additive and homogeneous functional.

The usual formulas for interpolation (polynomial or trigonometric), differentiation, and numerical integration, etc., are of the preceding form.

In applications it is important to be able to conveniently delimit the remainder. For this, at least in certain well-determined particular cases, attempts have been made to put the remainder in various convenient forms. For example, it has been obtained $R[f]$ in the form of a given linear combination of one or more values of one or more derivatives of certain orders of the function $f$ . The remainder was also expressed in the form of a definite integral. It is sufficient to quote Taylor's formula which gives an approximation of the value of the function $f$ for a given value of $x$ and whose remainder is given by the well-known Lagrange formula or by a well-known integral representation [4].

Much research has been done on the rest. We will content ourselves with citing the works of AA Markov [6]. GD Birkhoff [1], G. Kowalewski [5], R. v. Mises [7], J. Radon [21], E. Ya. Remez [22], A. Sard [23].

⁰⁰footnotetext: *) This work is also published in French in the journal "Mathematica" vol. 1 (24), fascicula 1.

In this paper we will highlight another expression of the remainder, which is more general, in the sense that, in general, it does not require the existence of derivatives, other than those that actually intervene in the formula $(*)$ .

The new shape we give to the remainder makes its structure stand out better. We obtained this result with the help of the theory of higher-order convex functions, which we studied elsewhere [12, 13].

Under a certain particular hypothesis made on the functions $f_{i}$ , a case that nevertheless encompasses a vast field of applications, the expression we find for the remainder is closely related to certain average formulas.

We will make some considerations on these average formulas and thus we will find part of DV Widder's results [28].

In this case it is easy to deduce the remainder expressed as a linear combination of derivatives, if, of course, these derivatives exist.

We obtained some of these results [16] in the particular case when the functions $f_{i}$ are reduced to successive powers $x^{i},i=0,1,\ldots,n$ his/hers $x$ , so in the case where the remainder cancels out for any polynomial of degree $n$ In this case, we also gave applications to certain formulas for derivation [17] and numerical integration [19].

This paper is divided into 4 parts. In § 1 we study the new expression of the remainder in the case when it has the form that we agree to call simple. In § 2 we study the average formulas that we have indicated. In § 3 we give examples for certain criteria that allow us to decide whether the remainder is of simple form or not. Finally, in § 4 we say a few words about the case when the remainder is not of simple form and we conclude this paragraph with an application. The results $\S\xi 3.4$ shows us, on the one hand, their connection with other known results, in particular with the results of E. Ya Remez [22] and, on the other hand, the degree of generality of the expression obtained for the remainder.

§ 1.

1.

All functions considered in this paper will be assumed to be real and of a real variable. We will denote by $AND$ the definition set of the function or the definition set of functions considered simultaneously. We will always specify, if necessary, the structure of $AND$ .

We denote with

V\left(\begin{array}[]{ll}g_{1},&g_{2},\ldots,g_{m}\\ x_{1},&x_{2},\ldots,x_{m}\end{array}\right)=\left|g_{j}\left(x_{i}\right)\right|_{i,j=1,2,\ldots,m}

determinant of function values

g_{1},g_{2},\ldots,g_{m}

(2)

on the points $x_{i}\in E,i=1,2,\ldots,m$ In the determinant (1), $g_{j}\left(x_{i}\right)$ is the element in line a $i$ -a and column a $j$ -a.

The determinant is obviously zero if the points $x_{i}$ or if the functions (2) are not distinct.

We will keep the notation (1) only for the case when the points $x_{i}$ are distinct. Otherwise, we will suitably modify the definition of the determinant (1). This modification consists in replacing the lines corresponding to each group of points $x_{i}$ confused by lines formed by the values of the functions (2) and their successive derivatives on these points. More precisely, either $z_{1},z_{2},\ldots,z_{p}$ the distinct points with which they coincide respectively $k_{1},k_{2},\ldots,k_{p}\left(k_{1},k_{2},\ldots,k_{p}\geqq 1,k_{1}+k_{2}+\ldots+k_{p}=m\right)$ between the points $x_{i}$ . Then, for each $i=1,2,\ldots,p$ , there exists exactly $k_{i}$ lines formed by the values of the functions (2) and their primes $k_{i}-1$ derivatives on the point $z_{i}$ This implies, of course, the existence of the derivatives considered. The number $k_{i}$ is the multiplicity order of the point $z_{i}$ .

Conveniently ordering the points $x_{i}$ , we can write the determinant (1) thus modified by

V(\underbrace{z_{1},z_{1},\ldots,z_{1}}_{k_{1}},\underbrace{g_{1},g_{2},\ldots,g_{m}}_{k_{2}}z_{2},z_{2},\ldots,z_{2},\ldots,\underbrace{z_{p},z_{p},\ldots,z_{p}}_{k_{p}}),

(3)

which is of the order $m=k_{1}+k_{2}+\ldots+k_{p}$ and in which $g_{s}^{(r-1)}\left(z_{i}\right)$ is the element in a $\left(k_{1}+k_{2}+\ldots+k_{i-1}+r\right)$ -th line and $s$ -th column, $v=1,2,\ldots,k_{i},i=1,2,\ldots,p\left(k_{1}+k_{2}+\ldots+k_{i-1}\right.$ is replaced by 0 if $i=1$ ).

We highlight the following particular cases
$1^{\circ}$ If $g_{i}=x^{i-1},i=1,2,\ldots,m$ , we will denote the determinant (1) by $V\left(x_{1},x_{2},\ldots,x_{m}\right)$ This is the Vandermonde determinant of the numbers 1. $x_{1},x_{2},\ldots,x_{m}$ and we have

V\left(x_{1},x_{2},\ldots,x_{m}\right)\stackrel{{\scriptstyle 1,2,\ldots,m}}{{=}}\prod_{i<j}\left(x_{j}-x_{i}\right),\quad\left(V\left(x_{1}\right)=1\right)

(4)

Also in this case the determinant (3) will be denoted by
$V(\underbrace{z_{1},z_{1},\ldots,z_{1}}_{k_{1}},\underbrace{z_{2},\ldots,z_{2}}_{k_{2}},\ldots,\underbrace{z_{p},z_{p},\ldots,z_{p}}_{k_{p}})$ where we assume that the points $z_{i},i=1,2,\ldots,p$ , are distinct.
$2^{\circ}$ In the case when all the points $x_{i}$ coincide with $x$ , we denote the determinant (1) modified with $W\left(g_{1},g_{2},\ldots,g_{m}\right)$ . This is the Wronskian of the functions (2). We therefore have

V(\underbrace{x,x,\ldots,x}_{m})=W\left(1,x,x^{2},\ldots,x^{m-1}\right)=(m-1)!!

where did I put $\alpha!!=1!2!\ldots\alpha!(0!!=1)$ .
2. We can also obtain the determinant (3) by passing the first limit if all the derivatives involved are continuous on $E$ , or at least in the vicinity of the points $z_{i}$ .

Whether $m$ distinct points $x_{j}^{(i)},j=1,2,\ldots,k_{i},i=1,2,\ldots,p$ and form the determinant $D$ of the order $m$ whose element of a $\left(k_{1}+k_{2}+\right.\left.+\ldots+k_{i-1}+\gamma\right)-a$ line and $s-a$ column is the difference divided (ordinary) by the order $r-1,\left[x_{1}^{(i)},x_{2}^{(i)},\ldots,x_{r}^{(i)};g_{s}\right],r=1,2,\ldots,k_{i},s==1,2,\ldots,m$ If we observe that this divided difference tends towards $\frac{1}{(r-1)!}g_{s}^{(r-1)}\left(z_{i}\right)$ when the points $x_{j}^{(i)},j=1,2,\ldots,r$ tend towards $z_{i}$ , we see that the determinant $D$ tends to the determinant (3) divided by $\prod_{i=1}^{n}\left(k_{i}-1\right)$ ! !, when $x_{j}^{(i)}\rightarrow z_{i},j=1,2,\ldots,k_{i},i=1,2,\ldots,p$ . Finally, if we multiply the determinant $D$ through the product

\prod_{i=1}^{p}V\left(x_{1}^{(i)},x_{2}^{(i)},\ldots,x_{k_{i}}^{(i)}\right)

(5)

and once we do a few elementary operations on the lines, we obtain the determinant

V\binom{g_{1},g_{2},\ldots\ldots,g_{n}}{x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}}^{(1)},x_{1}^{(2)},x_{2}^{(2)},\ldots,x_{k_{2}}^{(2)},\ldots,x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p}}^{(p)}}.

(6)

It follows that the determinant (3) is obtained by multiplying (6) by $\left(k_{i}-1\right)$ ! !, dividing 1 by (5) and then making the points $x_{j}^{(i)}$ to tend towards $z_{i}$ for $j=1,2,\ldots,k_{i},\quad i=1,2,\ldots,p$ .

The procedure of passing to the limit by which determinant (3) was obtained starting from determinant (1) can be generalized. Namely, a determinant of the form (3) can be obtained in the same way starting from determinants of the same form. We do not insist on this generalization because it will not be used in what follows.

As a first application we find the formula

	$\displaystyle V(\underbrace{z_{1},z_{1},\ldots,z_{1}}_{k_{1}},\underbrace{z_{2},z_{2},\ldots,z_{2}}_{k_{2}},\ldots,\underbrace{z_{p},z_{p},\ldots,z_{p}}_{k_{p}})=$
	$\displaystyle=\left[\prod_{i=1}^{p}\left(k_{i}-1\right)!!\right]^{1,2,\ldots,p}\left(\bar{z}_{j}-z_{i}\right)^{k_{i}k_{j}}$		(7)

We have, based on the well-known formula (see, e.g., IV Gonciarov [3]),

	$\displaystyle V\binom{1,\cos x,\sin x,\cos 2x,\sin 2x,\ldots,\cos mx,\sin mx}{x_{1},x_{2},\ldots\ldots,x_{2m+1}}=$
	$\displaystyle=2^{-m\prod_{i<j}^{1,2,\ldots 2m+1}}\left(2\sin\frac{x_{j}-x_{i}}{2}\right)$		(8)

from which it results and

	$\displaystyle V\binom{1,\cos x,\sin x,\cos 2x,\sin 2x,\ldots,\cos mx,\sin mx}{z_{1},z_{1},\ldots,z_{1},\underbrace{z_{2},z_{2},\ldots,z_{2}}_{k_{1}},\ldots,\underbrace{z_{p},z_{p},\ldots,z_{p}}_{k_{2}}}=$		(9)
	$\displaystyle=2^{-m}\left[\prod_{i=1}^{n}\left(k_{i}-1\right)!!\right]_{i<j}^{1,2,\ldots,p}\left(2\sin\frac{z_{j}-z_{i}}{2}\right)^{k_{i}k_{j}},\left(k_{1}+k_{2}+\ldots+k_{p}=2m+1\right)$

If in the following we consider a determinant (1) with the points $\dot{x}_{i}$ not all distinct, we will consider it modified in the manner explained above.
3. If among the points $x_{i}$ , on which the determinant (1) or the modified determinant (3) is defined, there exists one that has the order of multiplicity $k$ respectively an order of multiplicity $\leqq k$ , we will say that this point is repeated by $k$ , or respectively, it is repeated at most $k$ times.

Definition 1. - We will say that the functions (2) form an interpolation system or a system (I) on the set $E$ (having at least m points) if. we have

V\left(\begin{array}[]{lll}g_{1},&g_{2},&\ldots,\\ x_{1},&x_{2},&\ldots,\end{array}x_{m}\right)\neq 0

for any group of $m$ distinct points $x_{i}\in E,i=1,2,\ldots,m$
The property of forming a system (I) on $E$ , for the functions (2), is more restrictive than their linear-independence property (on $E$ ). In other words, any system (I) is formed by linearly independent functions but not every system of linearly independent functions forms an (I) system.

It is also of interest to complete Definition 1 by
Definition 2. - We will say that the functions (2) form a regular system (I) of order $k(1\leqq k\leqq m)$ on $E$ if we have (10) for any group of $m$ puncture $x_{i}\in E,i=1,2,\ldots,m$ , each repeating at most $k$ times.

If $k=m$ , we will say that the functions (2) form a complete regular system (I) (on $E$ ).

Regularity of the order $k$ so it means that the determinant (3) is $\neq 0$ if $z_{i}\in E,1\leqq k_{i}\leqq k,i=1,2,\ldots,p,k_{1}+k_{2}+\ldots+k_{p}=m$ , the points $z_{i}$ being distinct.

In definition 2 we always assume that if $k>1$ , derivatives of the order $k-1$ of the functions (2) are continuous on $E$ In this way, the regularity of the order $k>1$ implies continuity on $E$ of derivatives of the order $k-1$ of the functions (2). The assumption made previously allows us to avoid any difficulty. This is obviously a restriction but, according to TJ Stieltjes [26], it ensures the validity of the passage to the limit from no. 2.

One can obviously define the modified determinant ( 3 ), assuming more general differentiability conditions, from which also a more general notion of regularity results, but then the limit properties are more complicated. We will systematically leave such generalizations aside.

The crowd $E$ it can be any. In what follows the crowd $E$ will generally be an interval. Then the notion of derivative is the one known from elementary analysis.

It is clear that regularity of the order $k$ implies regularity of any lower order and that complete regularity implies regularity of any order $\leqq m$ In particular, the notions of system (I) and regular system (I) of order 1 are equivalent.

Finally, regularity of the order $k$ is equivalent to one of the following properties:
$1^{\circ}$ For any group of $m$ points (counted by their multiplicity orders) $z_{i}\in E$ , respectively, by the orders $k_{i}$ of multiplicity, $i=1,2,\ldots,p$ , $k_{1}+k_{2}+\ldots+k_{p}=m$ and for any group of $m$ NUMBERS $y_{i}^{()},j==0,1,\ldots,k_{i}-1,i=1,2,\ldots,p$ , there is a linear combination of the functions (2) and a single $\varphi(x)$ for which $\varphi^{(f)}\left(z_{i}\right)=y_{i}^{(f)},j=0,1,\ldots,k_{i}-1$ , $i=1,2,\ldots,p$ .

If $f$ is a function such that $y_{i}^{(j)}=f^{(j)}\left(z_{i}\right),j=0,1,\ldots,k_{i}-1$ , $i=1,2,\ldots,p$ , we will denote this linear combination and with

L(f\mid x)=L(\underbrace{z_{1},z_{1},\ldots,z_{1}}_{k_{1}},z_{2,}\underbrace{}_{k_{2}},z_{2},\ldots,z_{2},\ldots,z_{k_{p}},z_{p},\ldots,z_{p};f\mid x).

(11)

If the functions (2) form a regular system (I) of order $k$ and if $k_{i}\leqq k,i=1,2,\ldots,p$ , the linear combination (11) is well-determined (and unique).

It is clear that if $f$ reduces to a linear combination of the functions (2) and if the previous condition is verified, we have $L(f\mid x)=f$ .
$2^{\circ}$ A linear combination of the functions (2) cannot be cancelled on $m$ points, each of which is repeated at most $k$ times, without being identically null.

It is said that a function is canceled by $k$ times on a point, if this function and its primes $k-1$ derivatives are zero at this point.

Formula (7) shows us that the functions $x^{i},i=0,1,\ldots,m-1$ , forms a completely regular (I) system for any natural number $m$ and on a certain crowd $E$ .

Also formula (9) shows us that the functions $1,\cos ix,\sin ix,i==1,2,\ldots,m$ , forms a completely regular system (I), for any natural number $m$ and on any interval that does not contain a closed subinterval of length $2\pi$ , so, in particular on the interval $[0,2\pi$ ), closed on the left and open on the right.
4. If the functions (2) are continuous, we can find more complete results. Thus, we have
Theorem 1. - If the functions (2) : $1^{\circ}$ . are continuous on the interval $E$ , $2^{\circ}$ . forms a regular system (I) of order $k$ on $E$ ,
the determinant (1) does not change sign, as long as the points $x_{i}$ , each of which is repeated at most $k$ times, do not change their relative order of magnitude (e.g., as long as the functions $g_{i}$ remain in the order indicated by the string (2) and the points $x_{i}$ remain in the ascending order of their indices, $x_{1}\leqq x_{2}\leqq\ldots\leqq x_{m}$ ).

According to the previous definitions, for $k>1$ (but not for $k=$ 1) condition $2^{\circ}$ of the statement implies the continuity of functions (2).

Let us first assume that $k=1$ . For the sake of demonstration, let us assume the opposite. We can then find the points

x_{1}^{\prime}<x_{2}^{\prime}<\ldots<x_{m}^{\prime},x_{1}^{\prime\prime}<x_{2}^{\prime\prime}<\ldots<x_{m}^{\prime\prime}

(12)

so that we have

V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m}^{\prime}}>0,\quad V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{m}^{\prime\prime}}<0.

(13)

points

x_{i}=\lambda x_{i}^{\prime\prime}+(1-\lambda)x_{i}^{\prime},\quad i=1,2,\ldots,m

(14)

remain distinct for $0\leqq\lambda\leqq 1$ and the determinant (1) is a function of $\lambda$ perfectly determined and continues on $[0,1]$ .

A well-known property of continuous functions shows us that there is a $\lambda,0<\lambda<1$ so that the determinant (1), where the points $x_{i}$ are given by (14), to be equal to zero. This is in contradiction with the hypothesis that the functions (2) form a system (I).

Let us also note that from (12) it follows that, for $0\leqq\lambda\leqq 1$ , points (14) also verify the inequalities $x_{1}<x_{2}<\ldots<x_{m}$ staying within a length range $\leqq\max\left(x_{m}^{\prime}-x_{1}^{\prime},x_{m}^{\prime\prime}-x_{1}^{\prime\prime}\right)$ If in addition $0<\lambda<1$ and $x_{1}^{\prime}\neq x_{1}^{\prime\prime},x_{m}^{\prime}\neq x_{m}^{\prime\prime}$ , the points $x_{i}$ are inside the smallest interval containing the points $x_{i}^{\prime},x_{i}^{\prime\prime}$ .

Let's assume now $k>1$ . For the sake of demonstration, let us assume the opposite again. We can then find the points $u_{1}\leqq u_{2}\leqq\ldots\leqq u_{m}$ , each of which is repeated at most $k$ times and points $v_{1}\leqq v_{2}\leqq\ldots\leqq v_{m}$ , each of which is also repeated at most $k$ times, so that

V\binom{\dot{g}_{1},g_{2},\ldots,g_{m}}{u_{1},u_{2},\ldots,u_{m}}>0,\quad V\binom{g_{1},g_{2},\ldots,g_{m}}{v_{1},v_{2},\ldots,v_{m}}<0.

(15)

We can then find the variable points (12), tending respectively to the points $u_{i}$ and $v_{i}$ , so that the product of the determinants (13), through some functions that remain positive, tends to the respective determinants (15). It follows that this time too it is possible to find the points (12) such that we have (13). The proof therefore returns to that of the previous case.

Theorem 1 is therefore completely proven.
5. The linear combination (11) can be written

L(f\mid x)=f(x)-\frac{V\binom{g_{1},g_{2},\ldots,g_{m},f}{x_{1},x_{2},\ldots,x_{m},x}}{V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1},x_{2},\ldots,x_{m}}}

(16)

where $x_{i}$ are the points $z_{i}$ , with their multiplicity orders, in some order. Formula (16) has a precise meaning if $x$ does not coincide with one of the points $x_{i}$ . Otherwise we agree to replace the second term of the second member by zero. This convention is necessary to avoid confusion with the definition of the determinant (1) in the case when the points $x_{i}$ they are not all distinct.

From formula (16) it follows

L(f\mid x)-L(g\mid x)=f(x)-g(x)-\frac{V\binom{g_{1},g_{1},\ldots,g_{m},f-g}{x_{1},x_{2},\ldots,x_{m},x}}{V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1},x_{2},\ldots,x_{m}}}

In particular, if the points $x_{i}$ are distinct and the functions (2) are continuous, we deduce the inequality

|L(f\mid x)-L(g\mid x)|\leqq M\max_{i=1,2,\ldots,m}\left(\left|f\left(x_{i}\right)-g\left(x_{i}\right)\right|\right)

(17)

where $M(>0)$ is the maximum, in the smallest closed interval containing the points $x_{i}$ , of the continuous function

\frac{\sum_{i=1}^{m}\left|V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+1},\ldots,x_{m},x}\right|}{\left|V\binom{g_{1},g_{2},\ldots,g_{m}}{x_{1},x_{2},\ldots,x_{m}}\right|}

where the determinants that intervene (in the numerator) are defined by the second member of formula (1).

We can easily generalize this result to the case where the points $x_{i}$ they are not distinct.

We then deduce
theorem 2. - If: $1^{\circ}$ . the functions (2) are continuous and form a system (I) on the interval $E,2^{\circ}$ . the linear combination $\varphi$ of these functions is canceled on $m-1$ distinct points $x_{i},i=1,2,\ldots,m-1$ , without being identically null on $E$ ,
function $\varphi$ (is continuous and) changes sign passing through a point $x_{i}$ (which does not coincide with an extremity of $E$ ).

It is assumed, of course, $m>1$ .
This property is well known. For completeness, we will give its proof.

Let us suppose that, contrary to the statement, $\varphi$ does not change sign passing through the point $x_{1}$ , which does not coincide with one of its extremities $E$ We can then find the points $x_{1}^{\prime},x_{1}^{\prime\prime}$ thus $\mathrm{ca}:1^{\circ}.x_{1}^{\prime}<x_{1}<x_{1}^{\prime\prime},2^{\circ}$ , none of the points $x_{i},i=2,3,\ldots,m-1\mathrm{nu}$ belongs to the closed interval $\left[x_{1}^{\prime},x_{1}^{\prime\prime}\right]$ , $3^{\circ}.\varphi\left(x_{1}^{\prime}\right)\varphi\left(x_{1}^{\prime\prime}\right)>0$ Let us consider inequality (17) relative to the points $x_{1}^{\prime},x_{1},x_{2},\ldots,x_{m-1}$ , to the function $\varphi$ and to the linear combination $\varphi_{1}$ of functions
(2) which takes the same values as $\varphi$ on the points $x_{1}^{\prime},x_{2},x_{3},\ldots,x_{n-1}$ and for which $\varphi_{1}\left(x_{1}\right)=-\varepsilon\operatorname{sg}\varphi\left(x_{1}^{\prime\prime}\right)$ , uncle $\varepsilon$ is a positive number $<\frac{1}{2M}\left|\varphi\left(x_{1}^{\prime\prime}\right)\right|$ We then have

\left|\varphi_{1}\left(x_{1}^{\prime\prime}\right)-\varphi\left(x_{1}^{\prime\prime}\right)\right|=\left|L\left(\varphi_{1}\mid x_{1}^{\prime\prime}\right)-L\left(\varphi\mid x_{1}^{\prime\prime}\right)\right|\leqq M\varepsilon<\frac{\left|\varphi\left(x_{1}^{\prime\prime}\right)\right|}{2}

It follows that sg $\varphi_{1}\left(x_{1}^{\prime\prime}\right)=\operatorname{sg}\varphi\left(x_{1}^{\prime\prime}\right)$
It is now seen that $\varphi_{1}$ , without being identical to null, cancels out at the points $x_{2},x_{3},\ldots,x_{m-1}$ and at least once more on each of the open intervals $\left(x_{1}^{\prime},x_{1}\right),\left(x_{1},x_{1}^{\prime\prime}\right)$ This is $\hat{in}$ contradiction with the fact that functions (2) form a system (I).

Theorem 2 is proved.
6. Suppose that the $n+2$ functions

f_{0},f_{1},\ldots,f_{n},f_{n+1}

(18)

are defined and form a system (I) on $E$ It is easy to see that then the first $n+1$ of these functions

f_{0},f_{1},\ldots,f_{n}

(19)

are linearly independent on $E$ We say [ 13
] that the function $f$ is convex, respectively concave with respect to the sequence (19) of functions, if

V\binom{f_{0},f_{1},\ldots,f_{n},f}{x_{1},x_{2},\ldots,x_{n+2}}>0\text{ resp. }<0

(20)

for any system $x_{1}<x_{2}<\ldots<x_{n+2}$ of $n+2$ his points $E$ If
the function $f$ is convex or concave with respect to the sequence (19), the terms of the sequence together with this function form a system (I) (on $E$ ). Conversely, if the functions (18) are continuous and form a system (I), the function $f_{n+1}$ and, in general, any one of these functions is convex or concave with respect to any series formed with the others $n+1$ functions.

In what follows we will assume that the integer $n$ is $\geqq 0$ .
The previous definition can also be given meaning in the case of $n=-1$ Then the series (19) disappears and the convexity, respectively the concavity of the function $f$ returns to its positivity and negativity respectively $E$ .

The notion of convexity thus introduced generalizes that of higher-order convexity (of the order $n$ ) [12], which is obtained in the particular case

f_{i}=x^{i},\quad i=0,1,\ldots,n

(21)

In this case the function

f_{n+1}=x^{n+1}

(

\prime

)

is convex with respect to the sequence of functions (21), the interval $E$ being any. 10 - Studies $\mathdollar 1$ mathematical research
7. The definition inequalities (20) are not symmetric with respect to the points $x_{i}$ and the distinction between convexity and concavity depends on the order in which the functions occur (19). This is the reason why in the definition I emphasized that convexity and concavity are relative to the sequence and not to the set of functions (1).

We observe that if $f$ is convex or concave, the function $-f$ is concave or convex respectively. The set of convex functions (or concave) with respect to the sequence (19) remains invariant or changes to the set of concave (or convex) functions by a permutation of the functions (19).

To remove these asymmetries we introduce the notation

\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=V\left(\begin{array}[]{lll}f_{0},&f_{1},&\ldots,\\ x_{1},&x_{2},&\ldots,\end{array}x_{n+2}\right):V\binom{f_{0},f_{1},\ldots,f_{n},f_{n+1}}{x_{1},x_{2},\ldots,x_{n+2}},

where we assume that the functions (18) form a system (I) and that the points $x_{i}$ are distinct. Then the expression (22) has a perfectly determined meaning and is symmetric with respect to the points $x_{i}$ In the particular case (21), (21') this expression reduces to the divided difference of the function $f$ on the nodes $x_{1},x_{2},\ldots,x_{n+2}$ We will continue to use the term divided difference for expression (22) and for the points $x_{i}$ the name of the nodes (of this divided difference or on which this divided difference is defined). In the notation (22) we omitted to highlight the functions (18) because we will never encounter two different systems (18) simultaneously in our considerations.

The divided differences thus defined enjoy some properties which are expressed by the formulas

	$\displaystyle{\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=\left\{\begin{array}[]{l}0,\quad i=0,1,\ldots,n\\ 1,\quad i=n+1,\end{array}\right.}$		(23)
	$\displaystyle{\left[x_{1},x_{2},\ldots,x_{n+2};\alpha f+\beta g\right]=\alpha\left[x_{1},x_{2},\ldots,x_{n+2};f\right]+}$
	$\displaystyle-\mid-\beta\left[x_{1},x_{2},\ldots,x_{n+2};g\right]$		(24)

whatever the functions $f,g$ , the constants $\alpha,\beta$ and the distinct points $x_{i}\in E$ , $i=1,2,\ldots,n+2$ . Formula (24) expresses the linearity property of the divided difference.
8. With the help of divided differences, the definition of convexity can be stated (in a somewhat more precise form) as follows:

Definition 3. - The function f is convex, non-concave, non-convex or concave with respect to the functions (19) if

\left[x_{1},x_{2},\ldots,x_{n+2};f\right]>0,\geqq 0,\leqq 0,\text{ resp. }<0

(25)

points $x_{i}\in E,i=1,2,\ldots,n+2$ being distinct and arbitrary.
It is seen that the definition is independent of the order of the functions (19) and that the distinction between convex and concave functions is specified by the choice of the function $f_{n+1}$ which is, ipso facto, convex. We will see below
, when studying the rest $R[f]$ , that the introduction of divided differences satisfies requirements that go far beyond our simple desire to restore the symmetry of certain formulas considered above.

Convexity (concavity) is a particular case of non-concavity (non-convexity). However, for what follows it is useful to make a clear distinction between non-concave (non-convex) functions in general and only convex (concave) functions.

If $f$ - is convex or non-concave, $f$ is concave respectively non-convex and reciprocal.

The linear combination with all positive and all negative coefficients, respectively, of a finite number (at least 1) of non-concave functions is non-concave, respectively non-convex. If at least one of the functions considered is convex, the linear combination considered is convex or concave, respectively.

The limit of a convergent sequence (on $E$ ) of non-concave (non-convex) functions is a non-concave (non-convex) function.

A function $f$ can be both non-concave and non-convex. The functions that verify this property are those and only those whose divided difference is zero on any group of $n+2$ his points $E$ For this property to be verified it is necessary and sufficient that $/$ to reduce to a linear combination of the functions (19). The condition is obviously sufficient. But it is also necessary. Indeed, since the functions (19) are linearly independent, there exists $n+1$ distinct points $x_{i},i=1,2,\ldots,n+1$ , so that $V\left(\begin{array}[]{ll}f_{0},&f_{1},\ldots,f_{n}\\ x_{1},&x_{2},\ldots,x_{n+1}\end{array}\right)\neq 0[20]$ We have $V\binom{f_{0},f_{1},\ldots,f_{n},f}{x_{1},x_{2},\ldots,x_{n+1},x}=0$ , for $x\in E$ , from which the property results.

Among the other properties of convex functions we point out
theorem 3. - If: $1^{\circ}$ . the functions (18) are continuous and form a system (I) on the interval $E,2^{\circ}$ . function $/$ is continuous but is neither convex nor concave on $E$ ,
can be found $n+2$ distinct points $x_{i}\in E,i=1,2,\ldots,n+2$ , so that we have $\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=0$ .

Indeed, if the function nut is neither convex nor concave, then it is either non-concave or non-convex and then the property is obvious or two groups of each can be found $n+2$ distinct points $x_{i}^{\prime}\in E$ and $x_{i}^{\prime\prime}\in E$ , $i=1,2,\ldots,n+2$ , so that the divided differences

\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right],\left[x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{n+2}^{\prime\prime};f\right]

(26)

be different from zero and of opposite signs. It is then sufficient to apply Theorem 1, taking into account the definition formula (22) of divided differences.

We also deduce the more general property expressed by
Theorem 4. - If: $1^{\circ}$ . the functions (18) are continuous and form a system (I) on the interval $E,2^{\circ}$ . function $f$ is continuous on $E,3^{\circ}$ . $C$ is a number between the values $A,B$ of the divided differences (26),
one can find $n+2$ distinct points $x_{i}\in E;i=1,2,\ldots,n+2$ , so that we have $\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=C$ .

If $C$ coincides with $A$ or with $B$ , which necessarily occurs if $A=B$ , the property is obvious. Otherwise we have $(A-B)(B-C)<0$ . Taking into account (23), (24) it is easy to verify that the function $f-Cf_{n+1}$ is neither convex nor concave. It is then sufficient to apply Theorem 3 to this latter function.

From the observations made in the proof of Theorem 1, it follows that if $x_{1}^{\prime}<x_{2}^{\prime}<\ldots<x_{n+2}^{\prime},x_{1}^{\prime\prime}<x_{2}^{\prime\prime}<\ldots<x_{n+2}^{\prime\prime}$ , the points can be chosen $x_{i}$ , so that we have $x_{1}<x_{2}<\ldots<x_{n+2}$ and $x_{n+2}-x_{1}\leqq\max\left(x_{n+2}^{\prime}-x_{1}^{\prime},x_{n+2}^{\prime\prime}-x_{1}^{\prime\prime}\right)$ , and if $(A-C)(B-C)<0$ to have plus and minus $\left(x_{1}^{\prime},x_{1}^{\prime\prime}\right)<x_{i}<<\max\left(x_{n+1}^{\prime},x_{n+2}^{\prime\prime}\right),\quad i=1,2,\ldots,n+2$ .
9. If the functions (18) form a regular system (I) of order $k$ , we can take formula (22) to define any divided difference whose distinct nodes are repeated at most $k$ times. To highlight the multiplicity of nodes we will note this divided difference and with

[\underbrace{z_{1},z_{1},\ldots,z_{1}}_{h_{1}},\underbrace{z_{2},z_{2},\ldots,z_{2}}_{h_{2}},\ldots,\underbrace{z_{p},z_{p},\ldots,z_{p}}_{k_{p}};f]

(27)

where the nodes $z_{i}$ of the respective multiplicity orders $k_{i},i=1,2,\ldots,p$ , are distinct.

The results of 1a no. 2 show us that the divided difference (27) is the limit of the divided difference

\left[x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}}^{(1)},x_{1}^{(2)},x_{2}^{(2)},\ldots,x_{k_{2}}^{(2)},\ldots,x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p}}^{(p)};f\right]

on distinct nodes, if $x_{j}^{(i)}\rightarrow z_{i},j=1,2,\ldots,k_{i},i=1,2,\ldots,p$
In particular, assuming that the functions (18) form a completely regular system (I), we have

[\xi,\xi,\ldots,\xi;f]=\left[\frac{W\left(f_{0},f_{1},\ldots,f_{n},f\right)}{W\left(f_{0},f_{1},\ldots,f_{n},f_{n+1}\right)}\right]_{x=\xi}

(28)

The various properties of divided differences defined on distinct nodes can be extended to divided differences on not all distinct nodes, defined in the way shown above. For example, formulas (23), (24) obviously remain valid.

We observe that if the functions (18) form a completely regular system (I) and if the functions (19) are solutions (necessarily linearly independent) of the linear and homogeneous differential equation of order $n\neq 1$ ,

D[y]=y^{(n+1)}+\varphi_{1}(x)y^{(n)}+\ldots+\varphi_{n+1}(x)y=0

HAVE $W\left(f_{0},f_{1},\ldots,f_{n},f\right)=W\left(f_{0},f_{1},\ldots,f_{n}\right)D[f]$ and the formula (28) is clevin

[\xi,\xi,\ldots,\xi;f]=\left[\frac{D[f]}{D\left[f_{n+1}\right]}\right]_{\begin{subarray}{c}x=-\xi\\ i\end{subarray}}

(29)

The divided difference (27) exists, based on the given definition, only if the determinant $V$ from the numerator of the second member of formula (22) exists in the sense of no. 2. In the following we will assume that the function $f$ has all the intervening derivatives continuous. With this assumption the divided difference (27) exists under the above conditions.

More general divided differences can be defined on nodes not all distinct, by convenient limit crossings. These limit crossings can be done by means of the limits of ordinary divided differences (those corresponding to the particular case (21), (21')). In fact, this is how we proceed in this paper. One can also proceed directly, without going through the particular case (21), (21'). All these questions are closely related to the definition and existence of higher-order direct derivatives of a function.

To give an example, let us observe that in the particular case (21), (21'), the quotient (29) reduces to $\frac{1}{(n+1)!}f^{(n+1)}(\xi)$ and this result is valid, by virtue of our convention, if $f$ has a derivative of order $n+1$ continues, at least on the point $\xi$ . However, if for the first member of (29) (remaining in the particular case (21), (21')) we adopt as definition the limit of the divided difference $\left[x_{1},x_{2},\ldots,x_{n+1},\xi;f\right]$ when the points $x_{i},i=1,2,\ldots,n+1$ tend towards $\xi$ , formula (29) remains valid, as TJ Stieltjes [26] showed, only under the hypothesis of the existence of the derivative of order $n+1$ of the function at the point $\xi$ (function $f$ is assumed to be defined and bounded on $E$ ).

In what follows we will systematically leave aside such generalizations.
10. Let $R[f]$ a linear functional, defined on a vector space $\sqrt{f}$ made up of functions $f$ continue on the interval $E$ .

We assume that the functions (18) form a system (I) and belong to (f). In particular, they are continuous on $E$ .

If the linear functional $R[f]$ vanishes on the functions (19), it is zero on any linear combination of these functions. Such a functional is, for example,

K.\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};f\right]

(30)

where $K$ is a number independent of the function $f$ and $\xi_{i}$ saint $n+2$ distinct points of the interval $E$ .

We will now introduce
Definition 4. - We will say that the linear functional $R[f]$ , defined on $\mathbb{f}$ , is of simple form if, for any $f\in\mathbb{f}$ , it is of the form (30), where $K$ is a non-zero number, independent of the function $f$ , and $\xi_{i}$ saint $n+2$ distinctive points of $E$ (which may generally depend on the function $f$ ).

We then have
THEOREM 5. - The necessary and sufficient condition for the linear functional $R[f]$ to be of simple form, is to have $R[f]\neq 0$ for any function $f(\epsilon)$ convex with respect to the functions (19).

The condition is necessary. Indeed, if $R[f]$ is of the simple form. From (23) it follows first that $R\left[f_{n+1}\right]=K\neq 0$ From the formula

R[f]=R\left[f_{n+1}\right]\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};f\right]

(31)

it then follows that $R[f]\neq 0$ if $f$ is convex.
The condition is sufficient. If we have $R[f]\neq 0$ for any convex function, the same property is true for any concave function. Indeed, if $f$ is concave, the function $-f$ is convex and we have

R[f]=-R[-f]\neq 0

Whether $f\in\mathbb{F}$ and consider the auxiliary function

\varphi=R[f]\cdot f_{n+1}-R\left[f_{n+1}\right]\cdot f

(32)

we $\varphi\in\mathbb{F}$ and $R[\varphi]=0$ It follows that $\varphi$ is neither convex nor concave. By virtue of Theorem 3 we can find $n+2$ distinct points $\xi_{i}$ , $i=1,2,\ldots,n+2$ , so that we have $\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};\varphi\right]=0$ Taking into account (23), (24), from (32) we deduce formula (31).

Theorem 5 is therefore proven.
If $R[f]$ is of simple form, it cancels out on the functions (19). This property can be deduced directly from the fact that $R[f]\neq 0$ for any convex or concave function. To prove the property, let us assume the opposite, so that $R\left[f_{i}\right]\neq 0$ for a $i,0\leqq i\leqq n$ If we put $f=f_{i}$ in (32), we obtain a function $\varphi$ which is convex or concave. The equality $R[\varphi]=0$ is then in contradiction with the hypothesis.

An analogous demonstration shows us that if $R[f]\neq 0$ for any convex function, we have more precisely $R\left[f_{n+1}\right]R[f]>0$ for these functions. In other words $R[f]$ keeps its sign, which is the sign of $R\left[f_{n+1}\right]$ , for any convex function, so it keeps the opposite sign for any concave function.

It is also seen that if $R[f]$ is of simple form, we have $R\left[f_{n+1}\right]R[f]\geqq 0$ for any function $f$ non-concave and we have the opposite inequality for any non-convex function.

§ 2.

11.

REST $R[f]$ , in the case when it is of simple form, is expressed, by the formula (31), as a divided difference. The structure of the remainder therefore depends on the structure of the divided difference (22). The structure of this divided difference is specified by an important mean theorem due to DV Wid der [28]. This theorem takes place under an additional hypothesis made on the functions (19), an hypothesis which we will indicate below.

We will find DV Widder's results in a different way. Our results, which are sufficient for the study of the rest, are a little more general, but they only allow us to find part of DV Widder's results in the particular case examined by this author.

The additional hypothesis we discussed above is that the functions (19) form a system (I). This is not a consequence of the fact that the functions (18) form a system (I) (see, e.g., the example given in no. 16). To avoid any difficulty, we will assume in the following that the functions (18) are continuous.
12. We will use the following formula

	$\displaystyle V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{2},x_{3},\ldots,x_{n+2}}V\binom{f_{0},f_{1},\ldots,f_{n},f}{f_{0},f_{1},\ldots,f_{n},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3}}=$
$\displaystyle=$	$\displaystyle V\binom{f_{1},\ldots,f_{n+3}}{x_{2},x_{3},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3}}V\binom{f_{0},f_{1},\ldots,f_{n},f}{x_{1},x_{2},\ldots,x_{n+2}}+$
$\displaystyle--V$	$\displaystyle\binom{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+2}}V\binom{f_{0},f_{1},\ldots,f_{n},f}{x_{2},x_{3},\ldots,x_{n+3}}.$	(33)

To prove this formula, let us consider the determinant of order $2n+3$

\left|a_{r,s}\right|\quad r,s=1,2,\ldots,2n+3

(34)

uncle $a_{r,s}$ is the element in line a $r$ -a and column a $s$ -a and where

		$\displaystyle a_{s,s}=\left\{\begin{array}[]{ll}f_{s-1}\left(x_{r}\right),&r=1,2,\ldots,n+3\\ 0,&r=n+4,n+5,\ldots,2n+3\end{array}\right\}s=2,\ldots,n+2$
		$\displaystyle f_{s-n-3}\left(x_{r}\right),$
		$\displaystyle a_{r,r}=$
		$\displaystyle\begin{array}[]{ll}f_{s-n-3}\left(x_{r-n-2}\right),&r=1,i,n+3\\ f_{s-n-3}\left(x_{r-n-1}\right),&r=n+4,n+i+2,\ldots,n+i+1\\ s=n+3,n+4,\ldots,2n+3\end{array}$

This determinant is equal to zero. To see this, it is enough to transform it, first adding the line a $(n+2+j)$ - that day $j$ -a for $j=2,3,\ldots,i-1$ and the line of $(n+1+j)$ - that day $j$ -a for $j=i+1$ , $i+2,\ldots,n+2$ and then subtracting its column from $(n+2+s)$ -a for $s=1,2,\ldots,n+1$ In this way all the elements located in the last $n+1$ columns and firsts $n+3$ lines become null.

If we expand the determinant (34) according to Laplace's formula after the first $n+2$ columns, we obtain formula (33).

Formula (33) is valid for $2\leqq i\leqq n+2$ It is easy to see how we can write it for $i=2$ and for $i=n+2$ .

If the points $x_{i},i=1,2,\ldots,n+3$ are distinct, considering (2), from formula (33) we deduce

	$\displaystyle{\left[x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3};f\right]=}$
	$\displaystyle=A\left[x_{1},x_{2},\ldots,x_{n+2};f\right]+B\left[x_{2},x_{3},\ldots,x_{n+3};f\right]$		(35)

where, taking into account the fact that the functions (19) form a system (I), we have

	$\displaystyle A=\frac{V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{2},x_{3},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3}}V\binom{f_{0},f_{1},\ldots,f_{n},f_{n+1}}{x_{1},x_{2},\ldots,x_{n+2}}}{V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{2},x_{3},\ldots,x_{n+2}}V\binom{f_{1},\ldots,f_{n},f_{n+1}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3}}}$		(36)
	$\displaystyle B=\frac{V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+2}}V\binom{f_{0},f_{1},\ldots,f_{n},f_{n+1}}{x_{2},x_{3},\ldots,x_{n+3}}}{f_{0},f_{1},\ldots,f_{n},f_{n+1}}$		(37)
	$\displaystyle V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{2},x_{3},\ldots,x_{n+2}}V\left(\begin{array}[]{c}x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3}\end{array}\right)$

If in (35) we put $f=f_{n+1}$ , we find $A+B=1$ . But if $x_{1}<x_{2}<\ldots<<x_{n+3}(1<i<n+3)$ , theorem 1 shows us that the coefficients $A,B$ , which are independent of the function $f$ , are positive. It follows that if $x_{1}<x_{2}<\ldots<x_{n+3}(1<i<n+3)$ , the divided difference $\left[x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3};f\right]$ , is a generalized arithmetic mean (with positive weights) of the divided differences $\left[x_{1},x_{2},\ldots,x_{n+2};f\right],\left[x_{2},x_{3},\ldots,x_{n+3};f\right]$ .

In particular, in the case of (21), (21') we find the formula for the average of the ordinary divided differences

\begin{gathered}{\left[x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+3};f\right]=}\\ =\frac{\left(x_{i}-x_{1}\right)\left[x_{1},x_{2},\ldots,x_{n+2};f\right]+\left(x_{n+3}-x_{i}\right)\left[x_{2},x_{3},\ldots,x_{n+3};f\right]}{x_{n+3}-x_{1}}.\end{gathered}

13.

From the formula (35) of the average we deduce the more general property expressed by

THEOREM 6. - If $x_{1}<x_{2}<\ldots<x_{m}$ saint $m\geq n+2$ his points $E$ , the divided difference $\left[x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n+2}};f\right]\quad\left(1=i_{1}<i_{2}<\ldots<\right.\left.<i_{n+2}=m\right)$ on $n+2$ between these points is a generalized arithmetic mean (with convenient positive weights) of the divided differences

\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right],\quad i=1,2,\ldots,m-n-1

(38)

each $n+2$ consecutive points in the sequence of points $x_{i}$ So
we have

\left[x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n+2}};t\right]=\sum_{i=1}^{m-n-1}A_{i}\left[x_{i},x_{i+1},\ldots,x_{i+n+1};t\right]

(39)

coefficients $A_{i}$ being positive, independent of function $f$ and of a sum equal to 1.

The proof presents no difficulty. It can be done exactly as in the particular case (21), (21') [14], by complete induction on the number $m$ of points $x_{i}$ The positivity of the coefficients is a consequence
of this proof if $x_{1}<x_{2}<\ldots<x_{m}$ (and the fact that $i_{1}=1$ , $i_{n+2}=m$ ).

In addition to the hypotheses of theorem 6, the following inequalities are also deduced:

	$\displaystyle\min_{i=1,2,\ldots,m-n-1}$	$\displaystyle\left(\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]\right)\leqq\left[x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n+2}};f\right]\leqq$
		$\displaystyle\leqq\max_{i=1,2,\ldots,m-n-1}\left(\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]\right)$		(40)

These equalities can only occur at the same time, namely if and only if the divided differences (38) have the same value. $C$ , so if and only if for the function $f-Cf_{n+1}$ , these divided differences are all zero. We know that for this it is necessary and sufficient that the function $f$ to depend linearly on the functions $f_{i},i=0,1,\ldots,n+1$ on the points $x_{i},i=1,2,\ldots,m$ .
14. The previous results allow us to demonstrate, under the same assumptions,

THEOREM 7. - If the function f is continuous on the interval $E$ and if $x_{i},i=1,2,\ldots,n+2$ saint $n+2$ distinctive points of $E$ , we can find, inside the smallest interval containing the points $x_{i}$ , a point $\xi$ so that in any neighborhood of this point there exists $n+2$ distinct points $x_{i}^{\prime}\in E$ , $i=1,2,\ldots,n+2$ for which we have the equality

\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right]

(41)

We will first demonstrate that in (41) we can choose the points $x_{i}^{\prime}$ inside the smallest interval containing the points $x_{i}$ and in an interval of length less than a positive number $\varepsilon$ some date.

We can assume from the outset that $x_{1}<x_{2}<\ldots<x_{n+2}(n\geqq 0)$ . We divide each of the intervals $\left[x_{j},x_{j+1}\right],j=1,2,\ldots,n+1$ in $m$ equal parts, $m$ being a natural number $>2$ and which checks the inequality

m>\frac{n+1}{\varepsilon}\max_{i=1,2,\ldots,n+1}\left(x_{j+1}-x_{j}\right).

(42)

Whether $y_{1}<y_{2}<\ldots<y_{(n+1)m+1}$ all the division points thus obtained. We therefore have $x_{i}=y_{(i-1)m+1},\quad i=1,2,\ldots,n+2$ and, finally, considering (42),

	$\displaystyle y_{i+n+1}-y_{i}\leq\frac{n+1}{m}\max_{j=1,2,\ldots,n+1}\left(x_{i+1}-x_{j}\right)<\varepsilon$		(43)
	$\displaystyle i=1,2,\ldots,(n+1)m-n$

Whether $\left[y_{r},y_{r+1},\ldots,y_{r+n+1};f\right]$ one of the smallest and $\left[y_{s},y_{s+1},\ldots,y_{s+n+1};/\right]$ one of the biggest differences divided $\left[y_{i},y_{i+1},\ldots,y_{i+n+1};f\right]$ , $i=1,2,\ldots,(n+1)m-n$ . Formula (40) gives us

	$\displaystyle{\left[y_{r},y_{r+1},\ldots,y_{r+n+1};f\right]\leqq\left[x_{1},x_{2},\ldots,x_{n+2};f\right]\leqq}$
	$\displaystyle\leqq\left[y_{s},y_{s+1},\ldots,y_{s+n+1};f\right]$		(44)

We will distinguish two cases:
Case 1. The equalities do not hold in (44). Then based on Theorem 4, for

\begin{gathered}\left.A=\left[y_{r},y_{r+1},\ldots,y_{r+n+1};f\right]\quad B=\mid y_{s},y_{s+1},\ldots,y_{s+n+1};f\right]\\ C=\left[x_{1},x_{2},\ldots,x_{n+2};f\right]\end{gathered}

the property results if we take into account the observation made during the proof of Theorem 1 and taking into account (43). The hypotheses of Theorem 1 are satisfied here.

Case 2. The inequalities (44) both become equalities. We then have $\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=\left[y_{2},y_{3},\ldots,y_{n+3};f\right]$ , where $x_{1}<y_{2}<y_{n+3}<x_{n+2}$ and the property still follows from (43).

It is now easy to prove the existence of the point $\xi$ The previous reasoning shows us that we can find the strings of $n+2$ puncture $x_{1}^{(j)}<x_{2}^{(j)}<\ldots<x_{n+2}^{(j)}$ , $j=1,2,\ldots$ so that, assuming $x_{1}<x_{2}<\ldots<x_{n+2}$ , let's have

\begin{gathered}{\left[x_{1},x_{2},\ldots,x_{n+2};f\right]=\left[x_{1}^{(j)},x_{2}^{(j)},\ldots,x_{n+2}^{(j)};f\right]}\\ j=0,1,\ldots;x_{i}^{(0)}=x_{i};i=1,2,\ldots,n+2\end{gathered}

and

x_{1}^{(j)}<x_{1}^{(j+1)},x_{n+2}^{(j+1)}<x_{n+2}^{(j)},x_{n+2}^{(j+1)}-x_{1}^{(j+1)}\leqq\frac{1}{2}\left(x_{n+2}^{(j)}-x_{1}^{(j)}\right),j=0,1,\ldots

The common point $\xi$ of closed intervals $\left[x_{1}^{(j)},x_{n+2}^{(j)}\right],j=1,2,\ldots$ check the property you are looking for.

It is seen that the point $\xi$ also enjoys the property that the points can always be found $x_{i}^{\prime}$ so that $\xi$ to be inside the smallest interval containing these points (we say that $\xi$ separate the points $\left.x_{i}^{\prime}\right)$ .

Proprietatea exprimată de teorema 7 , cel puțin în cazul particular (21), (21’), se datoreşte lui A. Cauchy [2]
15. Putem completa teorema 7, observînd că putem totdeauna alege punctele $x_{i}^{\prime}$ astfel ca ele să fie echidistante. Aplicînd proprietatea funcției $f-\left[x_{1},x_{2},\ldots,x_{n+2};f\right]f_{n+1}$ , se vede că este suficient să demonstrăm că dacă avem

\left[x_{1},x_{2},\ldots,x_{n+2};t\right]=0,\quad x_{1}<x_{2}<\ldots<x_{n+2}

(45)

putem găsi $n+2$ puncte echidistante $x_{i}^{\prime},i=1,2,\ldots,n+2$ , cuprinse în intervalu1 închis $\left[x_{1},x_{n+2}\right]$ astfel ca să avem (41).

Vom distinge două cazuri :
Cazul 1. Printre diferentele divizate pe noduri echidistante şi cuprinse în [ $x_{1},x_{n+2}$ ], există cel puțin una care este pozitivă și cel puțin una care este negativă. In acest caz proprietatea rezultă deoarece prin procedeul întrebuintat la demonstrarea teoremei l, se poate construi o diferentă divizată pe noduri echidistante și care să fie nulă.

Cazul 2. Toate diferenţele divizate pe noduri echidistante și cuprinse în $\left[x_{1},x_{n+2}\right]$ sînt de același semn. Vom arăta că atunci funcţia $f$ , presupusă
continuă, este neconcavă sau neconvexă pe $\left[x_{1},x_{n+2}\right]$ . Pentru fixarea ideilor, să presupunem că diferentele divizate pe noduri echidistante sînt toate $\geqq 0$ (sau toate $\leqq 0$ ). Din teorema 6 rezultă că toate diferentele divizate pe noduri care se divid rational (rapoartele mutuale ale distanțelor dintre noduri sînt rationale) sînt $\geqq 0$ (sau $\leqq 0$ ). Din continuitatea funcției $f$ rezultă atunci că toate diferențele divizate sînt $\geqq 0$ (sau $\leqq 0$ ). Funcția $f$ este deci neconcavă (sau neconvexă) pe $\left[x_{1},x_{n+2}\right]$ .

Proprietatea căutată rezultă atunci din
L ema 1. - Dacă funcția continuă $f$ este neconcavă pe intervalul $\left[x_{1},x_{n+2}\right]$ si dacă avem (45), toate diferentele divizate ale functiei pe noduri apartinînd lui $\left[x_{1},x_{n+2}\right]$ , sînt nule.

Pentru demonstrare să presupunem că proprietatea nu este adevărată. Există atunci puncte distincte $x_{i}^{\prime}$ astfel ca $\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right]>0$ . Reunirea mulțimilor de puncte $x_{i},x_{i}^{\prime},i=1,2,\ldots,m+2$ formează un şir de cel putin $n+3$ si cel mult $2n+4$ puncte distincte ale intervalului $\left[x_{1},x_{n+2}\right]$ . Aplicînd teorema 6, împreuna cu consecințele ei relative 1 a cazurile cînd egalitatea are loc în (40), succesiv şirurilor parti $x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};x_{1},x_{2},\ldots,x_{n+2}$ , se ajunge la o contradictic cu (45).

In fine, dacă tinem seamă de rezultatele lui D. V. W id d e r [28], putem afirma că egalitatea (41) poate fi realizată cu noduri $x_{i}^{\prime}$ echidistante, distanța $\delta$ a două noduri consecutive fiind suficient de mică. In cazul cînd intervalu1 $E\supset\left[x_{1},x_{n+2}\right]$ , teorema de medie a lui D. V. Widder afirmă că se poate realiza rezultatul precedent cu noduri $x_{i}^{\prime}$ echidistante pentru care distanta $\delta$ este mai mică decît un număr fix independent de funcția $f$ .
16. Înainte de a merge mai departe să observăm că teorema 7 poate să nu aibă loc clacă funcţiile (19) nu formează un sistem (I).

Să considerăm funcţiile $f_{i}=x^{i+1}i=0,1,\ldots,n$ pe un interval $E$ care contine punctul 0. Aceste functii nu formează un sistem (I). Funcția $f_{n+1}=1$ este convexă sau concavă (convexă dacă $n$ este impar şi concavă dacă $n$ este par), în sensul definiției nesimetrice a convexității. Avem $\left[x_{1},x_{2},\ldots,x_{n+2};x^{n+2}\right]=(-1)^{n+2}x_{1}x_{2},\ldots,x_{n+2}$ . Dacă deci pentru funcţia continuă $x^{n+2}$ avem egalitatea (41), unde unul dintre punctele $x_{i}$ coincide cu 0 , unul din punctele $x_{i}^{\prime}$ va coincide în mod necesar cu 0 . Rezultă ușor că teorema 7 nu se aplică.
17. Rezultatele acestui § se pot extinde și la cazul cînd nodurile nu sînt distincte.

Să presupunem că nu numai funcțille (18) dar și funcțiile (19) formează un sistem (I) regulat de ordinul $k$ .

Teorema 6 se poate extinde la cazul cînd punctele $x_{1}\leqq x_{2}\leqq\ldots\leqq x_{m}(m\geq n+2)$ nu sînt toate distincte si acelaşi punct se repeta cel mult de $k$ ori. Pentru cele ce urmează va fi destul să ne ocupăm de extensiunea formulei (35) şi vom arăta că această formulă rămîne valabilă dacă
$x_{1}\leqq x_{2}\leqq\ldots\leqq x_{n+3}$ , acelaşi punct repetindu-se cel mult de $k$ ori. Mai mult încă, coeficientii respectivi $A,B$ , de sumă egală cu 1 , rămîn independenți de funcţia $f$ și sint pozitivi dacă $x_{1}<x_{i}<x_{n+3}$ (ceea ce implică $1<i<n+3$ )

Formula căutată se scrie unde putem presupune $p\geqq 3$ şi avem $k_{r}^{\prime}=k_{r}^{\prime\prime}=k_{r}^{\prime\prime\prime}=k_{r},r=2,3,\ldots j-1,j+1,\ldots,p-1,2\leqq j\leqq p-1\quad($ dacă $p>3),\quad k_{1}^{\prime}=k_{1}^{\prime\prime}=k_{1}k_{j}^{\prime\prime}=k_{j}^{\prime\prime\prime}=k_{j},\quad k_{p}^{\prime}=k_{p}^{\prime\prime\prime}=k_{p},\quad k_{1}^{\prime\prime\prime}=k_{1}-1,\quad k_{j}^{\prime}=k_{j}-1,k_{p}^{\prime\prime}=k_{p}-1$ ; $1\leqq k_{r}\leqq k,r=1,2,\ldots,p,k_{1}+k_{2}+\ldots+k_{p}=n+3$ .

Această formulă se obține din formulele (35) - (37), presupunînd $x_{1}<x_{2}<\ldots<x_{n+3}$ şi făcînd

	$\displaystyle x_{k_{1}+k_{2}+\ldots+k_{r-1}+s}=x_{s}^{(r)}\rightarrow z_{r},\quad s=1,2,\ldots,k_{r}\left(k_{0}=0\right)$		(47)
	$\displaystyle\gamma=1,2,\ldots,p$
	$\displaystyle z_{1}<z_{2}<\ldots<z_{p},\quad i=k_{1}+k_{2}+\ldots+k_{j}$

Se vede ușor cum trebuie modificată formula dacă $k_{1}=1,k_{j}=1$ sau $k_{p}=1$ .

Rezultă imediat că $A^{*},B^{*}$ sînt independenți de funcţia $f$ și că $A^{*}\geqq 0,B^{*}\geqq 0,A^{*}+B^{*}=1$ . Rămîne să se demonstreze că $A^{*}\neq 0$ , $B^{*}\neq 0$ . Pentru coeficientul $A^{*}$ acest lucru rezultă observînd că, cu ajutorul notatiilor (47), el se obtine din membrul al doilea al formulei (36) împărțind cei 4 determinanți (1) care figurează la numărător și la numitor prin expresia (5) ( $n=n+3$ ) multiplicată respectiv cu

		$\displaystyle\frac{V\left(x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}-1}^{(1)}\right)}{V\left(x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}}^{(1)}\right)}\cdot\frac{V\left(x_{1}^{(j)},x_{2}^{(j)},\ldots,x_{k_{1}-1}^{(j)}\right)}{V\left(x_{1}^{(j)},x_{2}^{(j)},\ldots,x_{k_{j}}^{(j)}\right)},\frac{V\left(x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p}-1}^{(p)}\right)}{V\left(x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p}}^{(p)}\right)}$
		$\displaystyle\frac{V\left(x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}-1}^{(1)}\right)}{V\left(x_{1}^{(1)},x_{2}^{(1)},\ldots,x_{k_{1}}^{(1)}\right)}\cdot\frac{V\left(x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p-1}}^{(p)}\right)}{V\left(x_{1}^{(p)},x_{2}^{(p)},\ldots,x_{k_{p}}^{(p)}\right)},\frac{V\left(x_{1}^{(j)},x_{2}^{(j)},\ldots,x_{k_{1}-1}^{(j)}\right)}{V\left(x_{1}^{(j)},x_{2}^{(j)},\ldots,x_{k_{j}}^{(j)}\right)}$

și trecînd la limită. Mai sus determinantii lui Vandermonde care nu au sens (pentru $k_{1}=1,k_{f}=1$ sau $k_{p}=1$ ) sînt înlocuiţi cu 1 . Dacă se efectuează aceste împărtiri, pe de o parte 1111 se schimbă valoarea coeficientului $A$ și, pe de altă parte, fiecare dintre determinanții (1) astfel împărțiți tinde către o limită bine determinată și diferită de zero. Rezultă că $A^{*}\neq 0$ . Se demonstrează în acelaşi fel că $B^{*}\neq 0$ . Demonstrația ne mai arată că
coeficientii $A^{*},B^{*}$ ai formulei (46) sînt bine determinați prin condiția ca să fie independenți de funcţia $f$ . Este uşor a se scrie valorile acestor coeficienti cu ajutorul determinanților (3).
18. Putem extinde teorema 7 la cazul cînd punctele $x_{i}$ nu sînt toate distincte. Intr-adevăr, presupunînd pe mai departe că funcţiile (18) și (19) sînt continue și formează cîte un sistem (I) regulat de ordinul $k$ , teorema 7 rămîne adevărată dacă printre punctele $x_{i}$ același punct se repetă cel mult. de $k$ ori.

Pentru a demonstra această proprietate, în virtutea chiar a teoremei 7, este sufficient să demonstrăm

L, e m a 2. - Dacă, pe lîngă ipotezele precedente, printre punctele $x_{i}$ există exact $p$ puncte distincte, cu $2\leqq p\leqq n+1$ ,
se pot găsi $n+2$ puncte $x_{i}^{\prime},i=1,2,\ldots,n+2$ , astfel ca : $1^{\circ}$ . fiecare se repetă cel mult de $k$ ori, $2^{\circ}$ . există printre ele cel putin $p+1$ distincte, $3^{\circ}$ . sînt cuprinse toate în cel mai mic interval închis care contine punctele $x_{i},4^{\circ}$ . egalitatea (41) este verificată.

Pentru simplificarea limbajului vom zice că o diferență divizată ale cărei noduri, aranjate în ordinea lor crescătoare, au succesiv ordinele de multiplicitate $k_{1},k_{2},\ldots,k_{p}\left(k_{1}+k_{2}+\ldots k_{p}=n+2\right)$ , este de tipul $\left(k_{1},k_{2},\ldots,k_{p}\right)$ . Condițiile $1^{\circ},2^{\circ}$ ale lemei însemnează că diferența divizată pe nodurile $x_{i}$ fiind de tipul ( $k_{1},k_{2},\ldots,k_{p}$ ), cu $1\leqq k_{i}\leqq k,i=1,2,\ldots,p$ , $2\leqq p\leqq n+1$ , se pot găsi punctele $x_{i}^{\prime}$ astfel ca diferența divizată pe aceste puncte să fie de tipul ( $k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{q}^{\prime}$ ), cu $1\leqq k_{i}^{\prime}\leqq k,i=1,2,\ldots,q$ , $q\geqq p+1$ .

Să considerăm deci diferența divizată pe nodurile $x_{i}$ și fie ( $k_{1},k_{2},\ldots,k_{p}$ ) tipul, iar $C$ valoarea acestei diferențe divizate. Să intercalăm între primele două noduri distincte un al ( $n+3$ )-lea nod, diferit de toate celelalte. Să aplicăm formula mediei (46) șirului de $n+3$ puncte astfel obținute, noul nod fiind acela care este eliminat în diferența divizată din membrul întîi. În membrul al doilea figurează diferentele divizate

\left[u_{1},u_{2},\ldots,u_{n+2};f\right],\left[v_{1},v_{2},\ldots,v_{n+2};f\right]

(48)

u_{1}\leqq u_{2}\leqq\ldots\leqq u_{n+2},v_{1}\leqq v_{2}\leqq\ldots\leqq v_{n+2},u_{1}<u_{n+2},v_{1}<v_{n+2},

care sînt respectiv de tipul $\left(k_{1},1,k_{2},k_{3},\ldots,k_{p-1},k_{p}-1\right)$ și $\left(k_{1}-1,1,k_{2},k_{3},\ldots,k_{p}\right)$ , unde trebuie suprimat $k_{1}-1$ dacă $k_{1}=1$ , şi $k_{p}-1$ dacă $k_{p}=1$ .

Trebuie acum să distingem trei cazuri :
Cazul 1. Diferentele divizate (48) au valori diferite. Atunci una are o valoare $A<C$ si cealaltă o valoare $B>C$ . Tinind seamă de felul cum - diferentă divizată (27) se obține ca limită de diferențe divizate pe noduri distincte, rezultă că putem găsi diferențele divizate

\left[u_{1}^{\prime},u_{2}^{\prime},\ldots,u_{n+2}^{\prime};f\right],\quad\left[v_{1}^{\prime},v_{2}^{\prime},\ldots,v_{n+2}^{\prime};f\right]

(49)

pe noduri distincte și ale căror valori sînt numerele $A^{\prime},B^{\prime}$ respectiv oricît de aproape de numerele $A,B$ , deci în particular, astfel ca $A^{\prime}<C<B^{\prime}$ .

Se poate uşor constata că putem chiar lua nodurile primei diferențe divizate (49) în intervalul ( $u_{1},u_{n+2}$ ) şi nodurile celei de a doua diferente divizate în intervalul ( $v_{1},v_{n+2}$ ). Aplicînd teorema 4 diferențelor divizate ( 49 ), putem găsi o diferență divizată avînd valoarea $C$ . Se vede că conditiile $2^{\circ},4^{\circ}$ ale lemei sînt verificate.

Cazul 2. Avem $k_{1}+k_{p}>2$ şi cele două diferențe divizate (48) sînt egale. Atunci ambele sînt egale cu $C$ şi sau prima (dacă $k_{p}>1$ ) sau a doua (dacă $k_{1}>1$ ) verifică condițiile $2^{\circ}$ și $4^{\circ}$ ale lemei.

Cazul 3. Avem $k_{1}=k_{p}=1$ și cele două diferenţe divizate (48) sînt egale cu $C$ . Avem atunci o diferență divizată egală cu $C$ şi de tipul $\left(1,1,k_{2},k_{3},\ldots,k_{p-1}\right)$ . Cu această diferență divizată se procedează în mod analog. Se vede atunci că dacă $k_{p-1}>1$ , cădem peste cazu1 1 sau 2 iar dacă $k_{p-1}=1$ se construieşte o diferenţă divizată egală cu $C$ și de tipul $\left(1,1,1,k_{2},k_{3},\ldots,k_{p-2}\right)$ . Deoarece cel puţin un $k_{i}$ este $>1$ , după un număr finit de operafii de acest fel se cade asupra cazului 1 sau 2 .

Astfel condiţiile $2^{\circ}$ și $4^{\circ}$ ale lemei sînt realizate. Să observăm că în timpul demonstrației, pe de o parte nu se întrece niciodată ordinul $k$ de multiplicitate şi, pe de altă parte, nu se iese niciodată din cel mai mic interval care conţine punctele $x_{i}$ . Deci și condițiile $1^{\circ}$ și $3^{\circ}$ ale lemei sint verificate.

Lema 2 este deci demonstrată.
Din cele ce preced rezultă și
teorema 8. - Dacă functilie (18) şi functiile (19) sînt continue si /ormează sisteme (I) regulate de ordinul $k$ pe intervalul $E$ si dacă functia $f$ este continuă și convexă, neconcavă, neconvexă resp. concavă in raport cu Junctiile (19),
prima, a doua, a treia respectiv a patra inegalitate (25) rămîne adevărată dacă nodurile $x_{i}$ nu sînt toate confundate si fiecare se repetă cel mult de $k$ ori.

Moreover, for non-concave functions and non-convex functions, the property results simply by passing to the limit and remains true if functions (18) and (19) form completely regular systems (I), even if the points $x_{i}$ they are all confused.

Theorem 8 results from the extension of Theorem 7 given in this issue
19. Theorem 7, extended in the above way, allows us to link the structure of a linear functional of simple form to the differential properties of the functions on which it is defined. Thus we have

THE $\mathrm{OREMA}9.-$ If: $1^{\circ}$ . the functionals (18) and (19) form completely regular systems (I) on the interval $E,2^{\circ}$ . linear functional $R[f]$ is of simple form, $3^{\circ}$ , function $f\in\mathbb{F}$ has a continuous derivative of order $n+1$ inside it $E$ ,
can be found, inside it $E$ , a point $\xi$ so that we have

F[f]=R\left[f_{n+1}\right][\xi,\xi,\ldots,\xi;f].

(50)

The proof follows immediately from Theorem 7 and the limit properties of divided differences with multiple nodes. The point $\xi$ is one of those that verifies theorem 7.

The difference divided by the second term of (50) can be calculated using formula (28) or formula (29).

We do not intend to delve into these issues further in this paper. We only recall that, in the particular case (21), (21'), we gave a generalization of Theorem 7 [18] which allows us to further specify the connection between the properties of the remainder $R[f]$ and the differential properties of different orders of the function $\not$ .

§ 3.

20.

In this § we will examine some criteria that allow us to decide whether a linear functional $R[f]$ is or is not of simple form. We will then make applications to the remaining few approximation formulas (*).

The linear combination (11) can be used to find an approximation formula of the form (*).

Whether $A[f]$ a linear functional defined on the vector space $(f$ formed by continuous functions defined on the interval $E$ and which have continuous derivatives on $E$ of all the orders that intervene. We will assume that the functions (18) and (19) belong to (f) and, to simplify matters, that they form completely regular systems (I). Moreover, for the validity of some of the results that follow, a regularity of an order lower than $n+2$ resp. $n+1$ is generally sufficient.

We will take as an approximation for $A[f]$ the defined and linear functional on $\widehat{t}$ ,

B[f]=A[L(f\mid x)],

(51)

where $L(f\mid x)$ is given by (11), relative to the functions (19).
This approximation procedure is well known and has been widely studied, especially in various particular cases.

L(f\mid x)=\sum_{i=1}^{p}\sum_{j=0}^{k-1}\varphi_{i,j}(x)f^{(j)}\left(z_{i}\right),\quad\left(f^{(0)}(x)=f(x),\quad\left(k_{1}+k_{2}+\cdots+k_{p}=n+1\right),\right.

points $z_{i},i=1,2,\ldots,p$ being distinct and $\varphi_{i,j},j=0,1,\ldots,k_{i}-1$ , $i=1,2,\ldots,p$ being well-determined linear combinations of the functions (19). We then have

B[f]=\sum_{i=1}^{p}\sum_{j=0}^{k_{i}-1}a_{i,j}f^{(j)}\left(z_{i}\right),

(52)

where $\left.a_{i,j}=A\mid\varphi_{i,j}\right],j=0,1,\ldots,k_{i}-1,i-1,2,\ldots,p$
There is an important particular case when the rest $R[f]$ of the approximation formula thus obtained is of simple form. We have, namely

THEOREM 10. - If: $1^{\circ}$ . linear functional $A[f]$ is positive, $2^{\circ}$ . orders of multiplicity $k_{i}$ of all points $z_{i}$ which are found $\hat{\imath}n$ inside the interval $E$ , are even,
the rest $R[f]$ of the approximation formula (*), constructed in the way shown above, is of simple form.

function $A[f]$ is positive if we have $A[f]\geqq 0$ , for any function $f$ (continuous) non-negative, the equality being true (if and) only if $f=0$ on $E$ .

Formula (16) gives us

f(x)-L(f\mid x)=\psi(x)\left[x_{1},x_{2},\ldots,x_{n+1},x;f\right]

(53)

points $x_{i}$ having the same meaning as in (16). In this formula we have

\psi(x)=V\binom{f_{0},f_{1},\ldots,f_{n},f_{n+1}}{x_{1},x_{2},\ldots,x_{n+1},x}:V\binom{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{n+1}}

if $x$ is different from one of the nodes $x_{i}$ .
Formula (53) is true for any $x\in E$ , provided that the second term is replaced by 0 if $x$ coincides with one of the nodes $x_{i}$ We have

R[f]=A[f-L(f\mid x)]=A\left[\psi\left[x_{1},x_{2},\ldots,x_{n+1},x;f\right]\right]

and the rest is of simple form because: $1^{\circ}$ . the divided difference appearing in the second member of formula (53) is, by virtue of theorem 8, positive if $f$ is a convex function, except at most $n+1$ points (points $x_{i}$ ) of his $E.2^{\circ}$ . the function is not identically zero and does not change sign on $E$ ; this property results from theorem 2 by taking the limit, $3^{\circ}$ . function $f-L(f\mid x)$ is continuous on $E$ It follows that this latter function is not identically zero and that it does not change sign on $E$ if $f$ is a convex function. Theorem 10 follows immediately.

The remainder is of the form (30) and if $(n+1)$ -derivative of $f$ exists and is continuous within it $E$ , of the same form as indicated in theorem 9. The constant $K=R\left[f_{n+1}\right]$ can also be calculated using the formula $K=R[\psi]$ , or using the formula $K=R[\psi+\varphi]$ , where $\varphi$ is a linear combination of the functions (19).

It is easy to generalize the previous result in the case when it is assumed that the functions (18) and (19) form regular (I) systems of order $k\geqq\max\left(k_{1},k_{2},\ldots,k_{p}\right)$ Finally, it is clear that an analogous property exists for a functional $A[f]$ negative, for which we therefore have $A[f]\leqq 0$ for any function $f$ nonnegative, the equality being true only for $j=0$ .

We note that many classical approximation formulas, for example so-called numerical (or mechanical) quadrature formulas, are of the previous form. We will recall some of these formulas below.
21. The well-known numerical quadrature formula

A[f]=\int_{0}^{2\pi}f(x)dx=\frac{2\pi}{m+1}\sum_{i=0}^{m}f\left(\frac{2i\pi}{m+1}\right)+R[f]

(54)

where $m$ is a natural number and $f$ a continuous function on the closed interval $[0,2\pi]$ , is of the previous form.

In this case $R[f]$ is null on the functions

f_{0}=1,f_{2i-1}=\cos ix,f_{2i}=\sin ix,\quad i=1,2,\ldots,m

(55)

to which the functions (19) now reduce. We have already shown that the functions (55) form a completely regular system (I) on the interval $[0,2\pi)$ This property is equivalent to the fact that a trigonometric polynomial of degree $m$ cannot have $2m+1$ distinct roots or not in the interval $[0,2\pi)$ , without being identically null.

Let us also consider the function

f_{2m+1}=x

(

\prime

)

Then the functions (55), (55') together also form a completely regular system (I) on $[0,2\pi)$ Indeed, a non-identical non-zero linear combination $\varphi$ of functions ( 55 ), ( $55^{\prime}$ ) cannot have more than $2m+1$ distinct roots or not in $[0,2\pi)$ Otherwise, the derivative $\varphi^{\prime}$ , which is a trigonometric polynomial of degree $m$ , would have at least $2m+1$ distinct roots or not in $[0,2\pi)$ It would follow that $\varphi^{\prime}=0$ , so that $\varphi$ is a constant $\neq 0$ , which is impossible.

Formula (54) is of the previous form. To obtain it, it is sufficient to take the function $L(f\mid x)$ (Lagrange-Hermite type trigonometric interpolation polynomial) relative to the simple node 0 and to the nodes $duble\frac{2i\pi}{m+1}$ , $i=1,2,\ldots,m$ It is easy to verify that (54) is the only formula of the form

\int_{0}^{2\pi}f(x)dx=Af(0)+\sum_{i=1}^{m}\left[\alpha_{i}f\left(\frac{2i\pi}{m+1}\right)+\beta_{i}f^{\prime}\left(\frac{2i\pi}{m+1}\right)\right]+R[f]

in which $A,\alpha_{i},\beta_{i}$ are independent of the function $f$ and the remainder of which cancels out on the functions (55).

The rest of formula (54) is of simple form and we have

R[t]=\frac{2\pi^{2}}{m+1}\left[\xi_{1},\xi_{2},\ldots,\xi_{2m+2};t\right]

function $f$ being continuous on $[0,2\pi]$ and having a continuous derivative on $(0,2\pi)$ . The points $\xi_{i}\in(0,2\pi),i=1,2,\ldots,2m+2$ are distinct.

If $f$ has a continuous derivative of order $2m+1$ on ( $0,2\pi$ ), we find the remainder given by J. Radon [21]. In our case

[\xi,\xi,\ldots,\xi;f]=\frac{1}{(m!)^{2}}\left[\frac{d}{dx}\left(\frac{d^{2}}{dx^{2}}+1^{2}\right)\left(\frac{d^{2}}{dx^{2}}+2^{2}\right)\cdots\left(\frac{d^{2}}{dx^{2}}+m^{2}\right)f\right]_{x\rightarrow\xi}

22.

Formula (54) is the trigonometric analogue of Gauss's classical numerical integration formula,

\int_{-1}^{1}f(x)dx=\sum_{i=1}^{m}\alpha_{i}f\left(\zeta_{i}\right)+R[t]

(56)

where $\zeta_{i},i=1,2,\ldots,m$ are the roots, all real, distinct and contained in ( $-1,1$ ), of the polynomial

P(x)=\frac{m!}{(2m)!}\cdot\frac{d^{m}}{dx^{m}}\left(x^{2}-1\right)^{m}

and whose remainder cancels out on any polynomial of degree $2m-1$ . Formula (56) is relative to the particular case (21), (21') and to obtain it it is sufficient to take the function $L(f\mid x)$ (LagrangeHermite interpolation polynomial) relative to double knots $\zeta_{i},i=1,2,\ldots,m$ By virtue of Theorem 10 the remainder is of simple form and we have ( $n=2m-1$ ),

R\left[x^{n+1}\right]=R\left[P^{2}\right]=\int_{-1}^{1}P^{2}dx=\frac{2^{2m+1}(m!)^{4}}{(2m+1)[(2m)!]^{2}}

The remainder is therefore of the form

R[f]=\frac{2^{2m+1}(m!)^{4}}{(2m+1)[(2m)!]^{2}}\left[\xi_{1},\xi_{2},\ldots,\xi_{2m+1};f\right]

(57)

the function being continuous on $[-1,1]$ and having a continuous derivative on $(-1,1)$ . The points $\xi_{i}\in(-1,1),i=1,2,\ldots,2m+1$ are distinct.

Existence and continuity of the derivative of the function $f$ in the study of the simplicity of the rest of the formulas (54) and (56) are imposed by the particular method by which we obtained this simplicity. It can be shown that the hypothesis of the existence of the derivative is superfluous, which we will effectively show below for Gauss's formula.
23. Let us consider a linear functional of the form

R[f]=\sum_{i=1}^{p}\sum_{\begin{subarray}{c}i=0\\ j=0\end{subarray}}^{k_{i}-1}c_{i,j}f^{(j)}\left(z_{i}\right),

(58)

where $n+2\geqq k_{1},k_{2},\ldots,k_{p}\geqq 1,k_{1}+k_{2}+\ldots+k_{p}=m\geqslant n+2,z_{1}<z_{2}<\ldots<z_{p}$ are points of the interval $E$ and $c_{i,j}$ , are independent coefficients of the function $f$ . Definition space $f$ of the functional is formed by the functions whose derivative of the max order $\left(k_{1}-1,k_{2}-1,\ldots,k_{p}-1\right)$ exists and is continuous on $E$ . We assume that the functions (18) and (19) belong to $\underset{f}{f}$ and form regular systems of max order $\left(k_{1},k_{2},\ldots,k_{p}\right)$ .

Whether $x_{1}\leqq x_{2}\leqq\ldots\leqq x_{m}$ points $z_{i}$ counted with their respective multiplicity orders. The functional (58) can also be written in the form

R[f]=R_{1}[f]+\sum_{i=1}^{m-1}\mu_{i}\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]

where $\mu_{i}$ are independent coefficients of the function $f$ .
$R_{1}[f]$ is an expression analogous to (58), but where only the values of the function appear $f$ and its successive derivatives on the first $n+1$ knots $x_{1},x_{2},\ldots,x_{n+1}$ (distinct or not). If one of these latter nodes is repeated by $k$ times, in $R_{1}[f]$ only the value of the function and its primes appear linearly (possibly with zero coefficients) $k-1$ derivatives at this point.

If we observe that in the divided divergence (27) (where $z_{i}$ are distinct) the coefficients of $f^{\left(k_{i}-1\right)}\left(z_{i}\right),i=1,2,\ldots,p$ are always different from zero, we see that the coefficients $\mu_{i}$ and the linear functional $R_{\mathbf{1}}[f]$ are completely determined by the linear functional (58).

For the linear functional (58) to be zero on the functions (19), it is necessary and sufficient that $R_{1}[f]$ be identically null. The condition is obviously sufficient (formula (23)). It is also necessary because the coefficients of can be successively nullified $R_{1}[f]$ , choosing for $f$ a convenient linear combination of functions (19).

From here the formula first results $R_{1}[f]=R\left[L\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right)\right]$ and then

Lemma 3. - For the linear functional (58) to be zero on the functions (19), it is necessary and sufficient that it be of the form

R[f]=\sum_{i=1}^{m-n-1}\mu_{i}\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]

(59)

where the coefficients $\mu_{i}$ are well-determined and independent of function $t$
From here we deduce
THEOREM 11. - If : $1^{\circ}$ . functions (18) and (19) form completely regular systems (I) on the interval $E,2^{\circ}$ . the linear functional (58) is zero on the functional (19), $3^{\circ}$ In expression (59) of this linear functional, the coefficients $\mu_{i}$ are of the same sign (all $\geqq 0$ or all $\leqq 0$ ), $4^{\circ}$ . assuming $x_{1}\leqq x_{2}\leqq\ldots\leqq x_{m}$ , we have

\sum_{i=1}^{m-1}\mu_{i}\left(x_{i+n+1}^{m}-x_{i}\right)\neq 0,

the linear functional (58) is of the simple form.
We assume here $m>n+2$ . The condition $4^{\circ}$ means that at least one of the coefficients $\mu_{i}$ is $\neq 0$ and at the same time the nodes $x_{i},x_{i+1},\ldots,x_{i+n+1}$ , corresponding to such a coefficient, are not all confused. The proof of Theorem 11 follows easily. Indeed, for a convex function all the terms of the sum (59) are of the same sign and at least one is $\neq 0$ .

The result is also valid for $m=n+2$ , suppressing the condition in the theorem $3^{\circ}$ .

It is easy to see that the condition $n+2\geqq k_{1},k_{2},\ldots,k_{p}$ is essential. In particular, this condition is satisfied by the linear functional (59). However, in the case when the condition is not satisfied, the linear functional (58) may not be of the indicated form and therefore Theorem 11 may not hold.
24. In the particular case (21), (21') we can give more complete results. In this case we can distinguish convexities from successive orders $n=-1,0,1,\ldots$ and the notion of simplicity of a linear functional is related to its degree of accuracy.

It is said that the linear functional $R[f]$ (or the corresponding approximation formula that has this remainder) has the degree of accuracy (integer) $n\geqq-1$ if $R\left[x^{i}\right]=0,i=0,1,\ldots,n,R\left[x^{n+1}\right]\neq 0$ Here we put $n=-1$ if $R[1]\neq 0$ and $n=\infty$ if $R\left[x^{i}\right]=0$ for $i=0,1,\ldots$ The degree of accuracy (finite or not) is always well determined. In what follows we consider only linear functionals having a finite degree of accuracy and which are defined, in particular, on any polynomial. For such a linear functional to have a finite degree of accuracy, it is necessary and sufficient that it is not zero on any polynomial. For example, the linear functional (58), assumed non-identically zero (more precisely with coefficients $c_{i,j}$ not all zero), has a finite degree of accuracy. Indeed, without restricting generality, it can be assumed that one of the coefficients $c_{i,k_{i}-1},i=1,2,\ldots,p$ is $\neq 0$ Either, for fixing ideas, $c_{r},k_{r-1}\neq 0$ . It can then be easily seen that $R\left[\frac{1}{x-z_{r}}\prod_{i=1}^{p}\left(x-z_{i}\right)^{k_{i}}\right]\neq 0$ .

For a linear functional to be of simple form, it is necessary for it to have a finite degree of accuracy.

We will prove
Theorem 12. - Assuming $x_{1}\leqq x_{2}\leqq\ldots\leqq x_{n+3}$ , because the linear junction

R\mid f]=\mu_{1}\left[x_{1},x_{2},\ldots,x_{n+2};f\right]+\mu_{2}\left[x_{2},x_{3},\ldots,x_{n+3};f\right],

(60)

(the coefficients $\mu_{1},\mu_{2}$ being independent of the function $f$ ) to be of simple form, it is necessary and sufficient that one of the conditions:
$1^{\circ}$ . The nodes $x_{i}$ they are not all confused and $\mu_{1}=-\mu_{2}\neq 0$ .
$2^{\circ}.\left(x_{n+2}-x_{1}\right)\mu_{1}+\left(x_{n+3}-x_{2}\right)\mu_{2}\neq 0,\quad\mu_{1}\mu_{2}\geqq 0,\quad$ to be verified.
From the condition $2^{\circ}$ it also follows that the nodes are not all confused. Moreover, if the first $n+2$ respectively the last $n+2$ nodes are confused, the coefficient $\mu_{2}$ respectively the coefficient $\mu_{1}$ is $\neq 0$ .

To prove the theorem it is necessary and sufficient to verify that in the cases $1^{\circ}$ and $2^{\circ}$ of the statement, the functional is of simple form, while in the other possible cases it is not of simple form. These possible cases are the following:
$3^{\circ}$ . The nodes $x_{i}$ they are all confused.
$4^{\circ}$ . The nodes $x_{i}$ are not all confused and $\mu_{1}\mu_{2}\geqq 0$ ,

\left(x_{n+2}-x_{1}\right)\mu_{1}+\left(x_{n+3}-x_{2}\right)\mu_{2}=0.

$5^{\circ}$ . The nodes $nu$ they are all confused and $\mu_{1}\mu_{2}<0,\mu_{1}+\mu_{2}\neq 0$
We will examine each of the 5 cases .
$1^{\circ}$ In this case expression (60) can be written as $\mu_{2}\left(x_{n+3}-x_{1}\right)\left[x_{1},x_{2},\ldots,x_{n+3};f\right]$ It is of the degree of accuracy $n+1$ and is of simple form, by virtue of Theorem 8.
$2^{\circ}$ The property follows from Theorem 11.
$3^{\circ}$ Based on the definition of the differences divided by not all distinct nodes, expression (60) is of the form $\left(\mu_{1}+\mu_{2}\right)[\underbrace{x_{1},x_{1},\ldots,x_{1}}_{n-2};f]$ Then the linear functional is: $3^{\prime o}$ . or identically null, so it is not of the simple form, $3^{\prime\prime}$ . Or has the degree of accuracy $n$ , but it cancels out on the function $\left|x-x_{1}\right|\left(x-x_{1}\right)^{n+1}$ which is convex of order $n$ , so it is not of simple form.
$4^{\circ}$ At least one of the coefficients $\mu_{1},\mu_{2}$ is zero and the linear functional (60) is: $4^{\prime\circ}$ . or identically null, $4^{\prime\prime\prime}$ or in the form of $3^{\circ}$ previous. In this case too, the functional is not of simple form.
$5^{\circ}$ The degree of accuracy is $n$ and we can write with $z_{1}<z_{2}<\ldots<z_{p}$ the distinct nodes, $k_{l}$ being the order of multiplicity of $z_{i}$ We have $1\leqq k_{1},k_{2},\ldots,k_{p}\leqq n+2,k_{1}+k_{2}+\ldots+k_{p}=n+3$ Let's consider the functions

\psi_{1}=-\left(\frac{x-\lambda_{1}-\left|x-\lambda_{1}\right|}{2}\right)^{n+2}\quad\psi_{2}=\left(\frac{x-\lambda_{2}+\left|x-\lambda_{2}\right|}{2}\right)^{n+2}

(61)

which are non-concave of the order $n$ and belong to the definition set $\mathbb{F}$ of the linear functional (60), as this set was defined in no. $\mathbf{23}$ Indeed, the functions (61) have (everywhere) continuous derivatives of order $n+1$ We will calculate on $R\left[\psi_{1}\right]$ and on $R\left[\psi_{2}\right]$ , assuming that $\lambda_{1}\in\left(z_{1},z_{2}\right)$ and $\lambda_{2}\in\left(z_{p-1},z_{p}\right)$ It is unnecessary to reproduce this calculation in detail here. We have

		$\displaystyle R\left[\psi_{1}\right]=\sum_{i=1}^{k_{1}-1}M_{i}\left(\lambda_{1}-z_{1}\right)^{n+2-i},\quad\left(z_{1}<\lambda_{1}<z_{2}\right)$
		$\displaystyle R\left[\psi_{2}\right]=\sum_{i=1}^{k_{p}-1}N_{i}\left(z_{p}-\lambda_{2}\right)^{n+2-i},\quad\left(z_{p-1}<\lambda_{2}<z_{p}\right),$

where

M_{k_{1}-1}=\frac{\binom{n+2}{k_{1}-1}\mu_{1}}{\left[\prod_{i=2}^{p-1}\left(z_{i}-z_{1}\right)^{k_{i}}\right]\left(z_{p}-z_{1}\right)^{k_{p-1}}},\quad N_{k_{p}-1}=\frac{\binom{n+2}{k_{p}-1}\mu_{2}}{\left[\prod_{l=2}^{p-1}\left(z_{p}-z_{i}\right)^{k_{i}}\right]\left(z_{p}-z_{1}\right)^{k_{1}-1}},

the other coefficients $M_{i},N_{i}$ , independent of $\lambda_{1}$ and $\lambda_{2}$ , having values that are unnecessary to calculate here.

We note that $M_{k_{1}-1},N_{k_{p}-1}$ are different from zero and of the same sign as $\mu_{1},\mu_{2}$ respectively. It is then seen that we can find a $\lambda_{1}$ close enough to $z_{1}$ and a $\lambda_{2}$ close enough to $z_{p}$ so that we have $R\left[\psi_{1}\right].R\left[\psi_{2}\right]<0$ . From an observation made in no. 10 it follows that the linear functional (60) cannot be of simple form.

Theorem 12 is completely proven.

The construction of functions (61) depends, to some extent, on the space (7. If this space is more restricted, e.g. if it contains only indefinitely differentiable functions on $E$ , we must replace the functions (61) by other convenient ones. We can avoid this modification by criteria analogous to those studied below (see no. 30).
25. Remaining in the particular case (21), (21'), if $R[f]$ is a linear functional defined on $\left(\mathcal{F},R^{*}[f]=R\left[f^{\prime}\right]\right.$ is a linear functional defined on the set $\mathscr{F}^{*}$ of continuous and differentiable functions whose derivative belongs to $\mathbb{F}$ It is easy to see that if $R[f]$ is of degree of accuracy $n(\geqq-1),R^{*}[f]$ is of degree of accuracy $n+1$ .

We also have
Theorem 13. - Under the previous assumptions, because $R[f]$ to be of simple form, it is necessary and sufficient that $R^{*}[f]$ to be of simple form.

The proof is immediate. It suffices to observe that the derivative of a convex function of order $n$ is a convex function of order $n-1$ and that all the primitives of such a function are convex functions of order $n+1$ .
26. To make an application, let's consider the numerical quadrature formula

\int_{a}^{b}f(x)dx=\sum_{i=0}^{k-1}\alpha_{i}f^{(i)}(a)+\sum_{i=0}^{l-1}\beta_{i}f^{(i)}(c)+\sum_{i=0}^{m-1}\gamma_{i}f^{(i)}(b)+R[f]

(62)

where $f$ is a continuous function on $[a,b]$ having the derivatives written continuous and $a<c<b$ .

Let us assume that the remainder of formula (62) is zero on any polynomial of degree $n-1=k+l+m-1\geq 0$ . Then the formula falls into the category of those studied at no. 20. The numbers $k,l,m$ can be zero, which means that the corresponding sum (hence the point $a,c$ or $b$ corresponding) does not occur in the second term of formula (62).

Particular cases of formula (62) have been studied by various authors and in particular by K. Petr [10, 11], GN Watson [27], N. Obreschkoff [9]. The method of these authors is different from the one presented here.

By virtue of Theorem 10, the remainder is of simple form if $l$ is even, in particular so if $l=0$ We will find this result below with the help of Theorems 12 and 13.

It is easily seen that $R[f]$ has a finite degree of accuracy which is equal to $n-1$ or with $n$ . Linear functional $R^{*}[f]=R\left[f^{\prime}\right]$ is of the form (58), with nodes not all confused, their total number being $n+2$ if $l=0$ and $n+3$ if $l>0$ We can now discuss the simplicity of the remainder with the help of Theorems 12 and 13.
$R[f]$ is of degree of accuracy $n$ if and only if

P(c)=\int_{a}^{b}(x-a)^{k}(x-c)^{l}(b-x)^{m}dx=0

(63)

This algebraic equation (of degree $l$ ) in $c$ has no real root in ( $a,b$ ) (in fact on the entire real axis) if $l$ is even, and has only one real root $c^{*}$ which is in $(a,b)$ if $l$ is odd. This result is obtained by noting that the derivative equation $P^{\prime}(c)=0$ is of the same shape. $R[f]$ is therefore of degree of accuracy $n$ if and only if $l$ is odd and $c=c^{*}$ .

Theorem 11 shows us that if $l=0,R^{*}[t]$ is of degree of accuracy $n$ and is of simple form. Therefore $R[f]$ is of degree of accuracy $n-1$ and of the simple form. It is also seen that if $l$ is odd and $c=c^{*}$ , it is of accuracy grade $n$ and it is of simple form.

To study the other possible cases, the coefficients must be calculated $\mu_{1},\mu_{2}$ of the formula (60) corresponding to $R^{*}[f]$ Some calculations, which we will not reproduce in detail, give us
$\mu_{1}=(-1)^{l+m}k!(c-a)^{l+1}(b-c)^{m}\alpha_{k-1},\mu_{2}=-m!(b-a)^{k}(b-c)^{l+1}\gamma_{m-1}$ , where

\begin{gathered}\alpha_{k-1}=\frac{(-1)^{l}}{(k-1)!(c-a)^{l}(b-a)^{m}}\int_{a}^{b}(x-a)^{k-1}(x-c)^{l}(b-x)^{m}dx,\quad(k>0)\\ \gamma_{m-1}=\frac{(-1)^{m-1}}{(m-1)!(b-a)^{k}(b-c)^{l}}\int_{a}^{b}(x-a)^{k}(x-c)^{l}(b-x)^{m-1}dx,\quad(m>0)\\ \alpha_{-1}=1,\gamma_{-1}=-1,\quad(0!=1)\end{gathered}

Applying Theorem 12, we see that if $l>0$ and if the rest $R[f]$ is of degree of accuracy $n-1$ , it is of simple form if and only if $\mu_{1}\mu_{2}>0$ This condition is checked if $l$ is an even number.

If $l$ is odd and $k>0$ , exists in $(a,b)$ a value $c_{1}$ his/her $c$ and only one for which $\mu_{1}=0$ and if $m>0$ a value $c_{2}$ his/her $c$ and only one for which $\mu_{2}=0$ .

we $c_{1}<c^{*}<c_{2}$ To prove the first inequality it is sufficient to observe that for the polynomial (63) we have

P(a)>0,\quad P\left(c_{1}\right)=\int_{a}^{b}(x-a)^{k-1}\left(x-c^{*}\right)^{l+1}(b-x)^{m}dx>0

The second inequality is proved in the same way.
It is immediately seen that if $c_{1}<c<c_{2}$ HAVE $\mu_{1}\mu_{2}<0$ , and if $c\leqq c_{1}$ or $c_{2}\leqq c$ HAVE $\mu_{1}\mu_{2}\geqq 0$ The results remain even if $k=0$ TAKING $c_{1}=a$ , and if $m=0$ taking then $c_{2}=b$ .

REST $R[f]$ of formula (62) is therefore of simple form only in the following three cases:
$1^{\circ}.l$ odd, $c=c^{*}$ .
$2^{\circ}$ . $l$ odd, $a<c\leqq c_{1}$ or $c_{2}\leqq c<b$ .
$3^{\circ}$ . $l$ about.

In case $1^{\circ}$ the rest has the form

R[f]=K^{*}\left[\xi_{1},\xi_{2},\ldots,\xi_{k+l+m+2};f\right]

and in cases $2^{\circ}$ and $3^{\circ}$ is of the form

R[f]=K\left[\xi_{1},\xi_{2},\ldots,\xi_{k+l+m+1};f\right]

where $\xi_{i}$ are, each time, distinct points of the interval ( $a,b$ ) and

K^{*}=\int_{a}^{b}(x-a)^{k}(x-c)^{l+1}(x-b)^{m}dx,\quad K=\int_{a}^{b}(x-a)^{k}(x-c)^{l}(x-b)^{m}dx

In the "symmetric" case $k=m$ , we have $c^{*}=\frac{1}{2}(a+b)$ and $c_{1}+c_{2}=a+b$
In the case of $l=1$ , we have

c_{1}=\frac{(m+1)a+kb}{m+k+1},c^{*}=\frac{(m+1)a+(k+1)b}{m+k+2},c_{2}=\frac{ma+(k+1)b}{m+k+1}

It can be shown that in all cases of simplicity of remainder, simplicity also occurs if the function is assumed to be continuous only on $[a,b]$ , having on the points $a,c,b$ the derivatives that actually appear in the second member of formula (62). The hypothesis of continuity of the derivative of the order $\max(k-1,l-1,m-1)$ was imposed only by the definition we adopted for differences divided by multiple nadas and by the criterion we relied on to demonstrate the simplicity of the remainder.
27. In the particular case (21), (21'), we will resume, specifying and completing it, a criterion we have already given [15].

Whether

\varphi_{n+1,\lambda}=\left(\frac{x-\lambda+|x-\lambda|}{2}\right)^{n}

(64)

where $n$ is a natural number. This is a non-concave function of order $n$ for anything $x$ Its derivative of the order $k$ exists if $0\leqq k\leqq n-1$ and is continuous for any $x$ We have, moreover,

\varphi_{n+1,\lambda}^{(k)}=\frac{n!}{(n-k)!}\varphi_{n+1-k,\lambda}\quad(0\leqq k\leqq n-1).

(65)

Whether $n$ a natural number and divide the finite and closed interval $[a,b]$ in $m>2n$ equal parts by points

\lambda_{i}=a+ih,i=0,1,\ldots,m,h=\frac{b-a}{m}\quad\left(\lambda_{0}=a,\lambda_{m}=b\right)

(66)

We denote with
$D_{j}^{i}[f]=\left[\lambda_{i},\lambda_{i+1},\ldots,\lambda_{i+j};f\right],\quad i=0,1,\ldots,m-j,j=0,1,\ldots,m$ (67) the divided (ordinary) differences of the function $/$ on (66) consecutive points,

Let's consider the functions

\psi_{m}=f_{m}+Q_{m}

(68)

where

	$\displaystyle f_{m}=(n+1)h\sum_{i=0}^{m-n-1}D_{n+1}^{i}[f]\varphi_{n+1,\lambda_{i+n}}$		(69)
	$\displaystyle Q_{m}=\frac{(-1)^{n}}{n!h^{n}}\left\{\sum_{r=0}^{n}(-1)^{r}f\left(\lambda_{r}\right)\left[\sum_{i=0}^{r}(-1)^{i}\binom{n+1}{r-i}\left(x-\lambda_{i+n}\right)^{n}\right]\right\}$		(70)

The function (68) is continuous and has a continuous derivative of order $n-1$ (so in any order $\leqq n-1$ ) for any $x$ It reduces to a polynomial of degree $n$ in each of the intervals $\left[\lambda_{i},\lambda_{i+1}\right],i=0,1,\ldots,m-1$ . $\Psi_{m}$ is what I once called an elementary function of order $n$ .

We have shown [15] that if $f$ is continuous on $\{a,b\}$ , the string $\left\{\psi_{m}\right\}$ converges uniformly over the entire interval $[a,b]$ by $f$ for $m\rightarrow{}^{c}$ We will complete this convergence property for the case when the function $f$ is differentiable a certain number of times.
28. Before stating and proving Theorem 14, which we will establish below, it is necessary to make some preliminary calculations.

Recurrence formula $\left.D_{j}^{i}[f]=\frac{1}{jh}\left\{D_{j-1}^{i+1}[f]-D_{j-1}^{i}[f]\right)\right\}$ allows us to establish various relations between the divided differences (67). Thus we have

(n+1)hD_{n+1}^{i}[f]=\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\sum_{j=0}^{n+1-k}(-1)^{j}\binom{n+1-k}{j}D_{k}^{i+j}|f|

(71)

Here $k$ is a whole such that $0\leq k\leq n+1$ For what follows it will suffice to assume that $0\leqq k\leqq n-1$ .

Taking into account formula (71), function (69) becomes

i_{m}=\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\sum_{r=0}^{m-k}D_{k}^{r}[f]\left[(-1)^{r}\sum_{i=r-n-1+k}^{r}(-1)^{i}\binom{n+1-k}{r-i}\varphi_{n+1,\lambda_{i+n}}\right]

(72)

where $\varphi_{n+1,\lambda_{i+n}}=0$ for $i<0$ and for $i>j$ if $x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right]$ , $j=-n,-n+1,\ldots,m-n-1$ .

To simplify, we introduce the notations

P_{i,u,v}=(-1)^{r}\sum_{i=u}^{u}(-1)^{i}\binom{n+1-k}{r-i}\left(x-\lambda_{i+n}\right)^{n}

(73)

Taking into account (73), we find

		$\displaystyle f_{m}=0,\text{ pentru }x\in\left[\lambda_{0},\lambda_{n}\right]$
		$\displaystyle f_{m}=\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\left[\sum_{r=0}^{n-k}D_{k}^{r}[f]P_{r,0,r}-\sum_{r=f+1}^{n-k}D_{k}^{r}[f]P_{r,j+1,r}+\right.$

$\left.+\sum_{r=n+1-k}^{n+1-k+j}D_{k}^{r}[j]P_{r,r-n-1+k,j}\right]$ , for $x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right]$ ,

		$\displaystyle j=1,\ldots,n-k-1,$
		$\displaystyle f_{m}=\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\left[\sum_{r=0}^{n-k}D_{k}^{r}[f]P_{k,0,r}+\sum_{r=n+1-k}^{2n-2k+1}D_{k}^{r}[f]P_{r,r-n-1+k,n-k}\right],$
		$\displaystyle\text{ pentru }x\in\left[\lambda_{2n-k},\lambda_{2n-k+1}\right],$
		$\displaystyle f_{m}=\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\left[\sum_{r=0}^{n-k}D_{k}^{r}[f]P_{r,0,r}+\sum_{r=n+1-k}^{j}D_{k}^{r}[f]P_{r,r-n-1+k,r+}\right.$
		$\displaystyle\left.+\sum_{k=j+1}^{n+1-k+j}D_{k}^{r}[f]P_{r,r-n-1+k}\right],\operatorname{pentru}x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right],$
		$\displaystyle j=n-k+1,n-k+2,\ldots,m-n-1.$

To put the polynomial (70) in a convenient form, we will apply the transformation formula

		$\displaystyle\quad\sum_{r=0}^{n}c_{r}f\left(\lambda_{r}\right)=(-1)^{k}k!h^{k}\sum_{r=0}^{n-k}D_{k}^{r}[f]\left[\sum_{s=0}^{r}\binom{k+s-1}{s}c_{r-s}\right]+$
		$\displaystyle+(-1)^{n}\sum_{r=n+1-k}^{n}(-1)^{r}(n-r)!h^{n-r-r}D_{n-r}^{r}[f]\left[\sum_{s=0}^{r}\binom{n-r+s}{s}c_{r-s}\right].$
		Let's take

c_{r}=(-1)^{r}\sum_{i=0}^{r}(-1)^{i}\binom{n+1}{r-i}\left(x-\lambda_{i+n}\right)^{n},r=0,1,\ldots,n

If we take into account the well-known formula (see, e.g., E. Netto [8])
we deduce

\sum_{s=0}^{l}(-1)^{s}\binom{s+a}{s}\binom{b}{t-s}=\binom{b-a-1}{t}

\begin{gathered}\sum_{s=0}^{r}\binom{k+s-1}{s}c_{r-s}=(-1)^{r}\sum_{i=0}^{r}(-1)^{i}\left[\sum_{s=0}^{r-i}(-1)^{s}\binom{k+s-1}{s}\binom{n+1}{r-i-s}\right]\left(x-\lambda_{i+n}\right)^{n}=\\ =P_{r,0,r}\\ \sum_{s=0}^{r}\binom{n-r+s}{s}c_{r-s}=(-1)^{r}\sum_{i=0}^{r}(-1)^{i}\binom{r}{r-i}\left(x-\lambda_{i+n}\right)^{n}\end{gathered}

where, finally,

	$\displaystyle Q_{m}=\frac{(-1)^{n-k}k!}{n!h^{n-k}}\sum_{r=0}^{n-k}D_{k}^{r}(f]P_{r,0,r}+$
	$\displaystyle+\sum_{r=n+1-k}^{n}\left\{\frac{(n-r)!}{n!h^{r}}D_{n-r}^{r}[f]\left[\sum_{i=0}^{r}(-1)^{i}\binom{r}{r-i}\left(x-\lambda_{i+n}\right)^{n}\right]\right\}.$		(74)

We will now calculate the derivatives of the function (68). Note that

\sum_{i=0}^{r}(-1)^{i}\binom{r}{r-i}\left(x-\lambda_{i+n}\right)^{n-k}=(-1)^{r}r!h^{r}D_{r}^{n}\left[(x-t)^{n-k}\right]

where we consider $x$ as a parameter and $t$ as the variable of the polynomial $(x-t)^{n-k}$ whose divided difference is calculated on the nodes $\lambda_{n},\lambda_{n+1},\ldots,\lambda_{n+r}$ . However, the difference divided by the order $r$ of a polynomial of degree $r-1$ is identically zero. It follows that the derivative of the order $k$ of the second sum of the second term of formula (74), vanishes. It is seen in the same way that $P_{r,r-n-1+k,r}^{(k)}=0$ for $r\geqq n+1-k$ .

So we have

	$\displaystyle\psi_{m}^{(k)}=$	$\displaystyle\frac{(-1)^{n-k}k!}{n!n^{n-k}}\sum_{r=0}^{n-k}D_{k}^{r}[f]P_{r,0,r}^{(k)}\quad\text{ pentru }x\in\left[\lambda_{0},\lambda_{n}\right]$
	$\displaystyle\psi_{m}^{(k)}=$	$\displaystyle\frac{(-1)^{n+1-k}k!}{n!h^{n-k}}\left[-\sum_{r=j+1}^{n-k}D_{h}^{r}[f]P_{r,j+1,r}^{(k)}+\sum_{r=n+1-k}^{n+1-k+i}D_{k}^{r}[/]P_{r,i-n-1+k,j}^{(k)}\right]$
		$\displaystyle\text{ pentru }x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right],j=1,\ldots,n-k-1$
	$\displaystyle\psi_{m}^{(k)}=$	$\displaystyle\frac{(-1)^{n+k-k}k!}{n!h^{n-k}}\sum_{r=j+1}^{n-k+j}D_{k}^{r}[f]P_{r,r-n-1+k,j}^{(k)}$
		$\displaystyle\text{ pentru }x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right],j=n-k,n-k+1,\ldots,m-n-1$

We will also need some convenient delimitations of derivatives of the order $k$ of the polynomials (73) that intervene in these formulas.

For $0\leqq s\leqq r\leqq n-k,x\in\left[\lambda_{0},\lambda_{2n-k}\right]$ HAVE

\begin{gathered}\left|P_{r,s,r}^{(k)}\right|\leqq\frac{n!}{(n-k)!}\sum_{i=0}^{r}\binom{n+1-k}{r-i}\left|x-\lambda_{i+n}\right|^{n-k}=\\ =\frac{n!}{(n-k)!}\sum_{i=0}^{r}\binom{n+1-k}{i}\left|x-\lambda_{r+n-i}\right|^{n-k}\leqq\\ \leqq\frac{n!}{(n-k)!}\sum_{i=0}^{r}\binom{n+1-k}{i}\left[\max_{x\in\left[\lambda_{0},\lambda_{2n-k}\right]}\left|x-\lambda_{r+n-i}\right|^{n-k}\right]\leqq\\ \leqq\frac{n!h^{n-k}}{(n-k)!}(2n-k)^{n-k}\left(2^{n+1-k}-1\right)\end{gathered}

So if we put

M=\frac{k!}{(n-k)!}(2n-k)^{n-k}\left(2^{n+1-k}-1\right)

(75)

we have, in particular,

\begin{gathered}\left|P_{r,0,r}^{(k)}\right|\leqq\frac{n!h^{n-k}}{k!}M,\quad\text{ pentru }0\leqq r\leqq n-k,\quad x\in\left[\lambda_{0},\lambda_{n}\right]\\ \left\lvert\,P_{r,j+1,r}^{(k)}\leqq\frac{n!h^{n-k}}{k!}M\right.,\quad\text{ pentru }j+1\leqq r\leqq n-k,\quad x\in\left[\lambda_{j+11},\lambda_{j+n+1}\right]\\ \quad j=0,1,\ldots,n-k-1\end{gathered}

For $j+1\leqq r\leqq n+1-k+j,x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right],j=0,1,\ldots;n-n-1$ ,
we have

		$\displaystyle\left\|P_{r,r-n-1+k,j}^{(k)}\right\|\leqq\frac{n!}{(n-k)!i=r-n-1+k}\binom{n+1-k}{r-i}\left\|x-\lambda_{i+n}\right\|^{n-k}\leqq$
		$\displaystyle\leqq\frac{n!}{(n-k)!}\sum_{i=0}^{n+1-k+j-r}\binom{n+1-k}{i}_{x\in\left[\lambda_{j+n^{\prime}}\lambda_{j+n+1}\right]}\left\|x-\lambda_{r+k-1+i}\right\|^{n-k}\leqq$
		$\displaystyle\leqq\frac{n!h^{n-k}n+1-k+j-r}{(n-k)!}\sum_{i=0}\binom{n+1-k}{i}(n+2-k+j-r-i)^{n-k}\leqq$
		$\displaystyle\leqq\frac{n!h^{n-k}}{(n-k)!}\left(n+2-k+j-r^{\prime}\right)^{n-k}\sum_{i=0}^{n+1-k+i-r}\binom{n+1-k}{i}<$
		$\displaystyle<\frac{n!h^{n-k}}{(n-k)!}(n+1-k)^{n-k}\left(2^{n+1-k}-1\right)\leqq\frac{n!h^{n-k}}{k!}M.$

Better delimitations can be found. I have given delimitations of this kind in another paper [15]. For what follows it is sufficient to note that the number (75) is independent of $m$ (and of $j$ ).
29. I will demonstrate the act.

THEOREM 14. - Given a natural number $n$ and the whole $k$ so that $0\leqq k\leqq n-1$ , if the function $\mid$ admits $o$ derived from the order $k$ continue at the intersection $I$ of the tinnitus and closed interval $[a,b]$ with an open interval,
the series of derivatives of order $k\left\{\psi_{m}^{(k)}\right\}$ , of the functions (68) converges uniformly to the derivative of order $k,f^{(l)}$ , of the function ! for $m\rightarrow\infty$ and on any closed subinterval of $I$ .

The zeroth derivative of a function coincides with the function itself.
The conclusion of the statement means that the convergence is uniform on $\left[a^{\prime},b^{\prime}\right]\subseteq[a,b]$ if $f^{(k)}$ is continuous on $\left[a^{\prime},b^{\prime}\right]$ and if, in addition, $a<a^{\prime}$ , is continuous and on an interval $\left[a^{\prime\prime},a^{\prime}\right)$ with $a<a^{\prime\prime}<a^{\prime}$ , and if $b^{\prime}<b$ is continuous and on an interval ( $b^{\prime},b^{\prime\prime}$ ] with $b^{\prime}<b^{\prime\prime}<b$ .

To demonstrate, we will delimit the difference $f^{(k)}-\psi_{m}^{(k)}$ .

If in his expression $\psi_{m}^{(k)}$ we replace all divided differences $D_{k}^{r}[f]$ by 1, the function $f_{m}^{(k)}$ the polynomial also cancels out identically $Q_{m}^{(k)}$ it reduces to

		$\displaystyle\frac{(-1)^{n-k}k!}{n!h^{n-k}}\sum_{r=0}^{n-k}P_{r,0,r}^{(k)}=$
	$\displaystyle=$	$\displaystyle\frac{(-1)^{n-k}k!}{(n-k)!h^{n-k}}\sum_{r=0}^{n-k}(-1)^{r}\left[\sum_{i=0}^{r}(1-)^{i}\binom{n+1-k}{r-i}\left(x-\lambda_{i+n}\right)^{n-k}\right]=$
	$\displaystyle=$	$\displaystyle\frac{(-1)^{n-k}k!}{(n-k)!h^{n-k}}\sum_{i=0}^{n-k}(-1)^{i}\left[\sum_{i=0}^{n-k}(-1)^{r}\binom{n+1-k}{r-i}\right]\left(x-\lambda_{i+n}\right)^{n-k}=$
	$\displaystyle=$	$\displaystyle\frac{k!}{(n-k)!h^{n-k}}\sum_{i=1}^{n-k}(-1)^{i}\binom{n-k}{i}\left(x-\lambda_{i+n}\right)^{n-k}=$
	$\displaystyle=$	$\displaystyle\frac{k!}{(n-k)!}\sum_{i=0}^{n-k}(-1)^{i}\binom{n-k}{i}(n-k-i)^{n-k}=k!$

In this calculation we have taken into account an observation already made on the divided differences of a polynomial. It is seen that the expression is independent of $x$ and so one can take (e.g.) $x=\lambda_{2n-k}$ .

It follows that the difference $f^{(k)}-\psi_{m}^{(k)}$ is obtained from $\psi_{m}^{(k)}$ replacing $D_{k}^{r}[f]\operatorname{prin}\frac{f^{(k)}}{k!}-D_{k}^{r}[f],\quad r=0,1,\ldots,m-k$ .

Taking into account the calculations made in the previous section, we have

\begin{gathered}\left|f^{(k)}-\psi_{m}^{(k)}\right|\leq M\sum_{r=0}^{n-k}\left|\frac{f^{(k)}}{k!}-D_{k}^{r}[j]\right|,\text{ pentru }x\in\left[\lambda_{0},\lambda_{n}\right]\\ \left|f^{(k)}-\psi_{m}^{(k)}\right|\leq M\sum_{r=j+1}^{n+1-k+j}\left|\frac{f^{(k)}}{k!}-D_{k}^{r}[j]\right|,\text{ pentru }x\in\left[\lambda_{y+n},\lambda_{j+n+1}\right]\\ j=0,1,\ldots,m-n-1\end{gathered}

Be it now $\left[a^{\prime},b^{\prime}\right]$ a closed subinterval of $I$ Let us first assume that $a<a^{\prime}<b^{\prime}<b$ and then $a<a^{\prime\prime}<a^{\prime},b^{\prime}<b^{\prime\prime}<b$ , derivative of the order $k,f^{(k)}$ being continuous on $\left[a^{\prime\prime},b^{\prime\prime}\right]$ Let us denote by $\omega_{k}(\delta)$ the oscillation modulus of $\frac{1}{k!}f^{(k)}$ on the interval $\left[a^{\prime\prime},b^{\prime\prime}\right]$ .

Let's take the natural number $m$ big enough for us to have

m>\max\left(2n,\frac{n(b-a)}{a^{\prime}-a^{\prime\prime}},-\frac{b-a}{b^{\prime\prime}-b^{\prime}},\frac{2(b-a)}{b^{\prime\prime}-a^{\prime}}\right)

(76)

and let's put

j_{0}=\left\lfloor\frac{a^{\prime}-a}{h}\right\rfloor-n,\quad j_{1}=\left\lfloor\frac{b^{\prime\prime}-a}{h}\right\rfloor-n-1,\quad h=\frac{b-a}{m},

where $\lfloor\alpha\rfloor$ means the largest integer less than or equal to $\alpha$ .

We then have $0\leq j_{0}+1,j_{0}\leqq j_{1}$ and $\left[a^{\prime},b^{\prime}\right]\subseteq\left[\lambda_{j_{0}+n},\lambda_{j_{1}+n+1}\right]\subseteq\subseteq\left[\lambda_{j_{0}+1},\lambda_{j_{1}+n+1}\right]\subseteq\left[\overline{a^{\prime\prime}},b^{\prime\prime}\right]$ .

If $j_{0}\leqq j\leqq j_{1},j+1\leqq r\leqq n+1-k+j$ , the nodes of the divided difference $D_{k}^{r}[f]$ I am in the range $\left[\lambda_{j+1},\lambda_{j+n+1}\right]\subseteq\left[a^{\prime\prime},b^{\prime\prime}\right]$ , where $f^{(k)}$ is continuous. There is then a point $\xi$ so that

D_{k}^{r}[t]=\frac{1}{k!}f^{(k)}(\xi),\quad\xi\in\left[\lambda_{j+1},\lambda_{j+n+1}\right]

and it turns out that

\left|\frac{f^{(k)}}{k!}-D_{k}^{r}[f]\right|\leqq\omega_{k}(nh),\text{ pentru }x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right].

So we have

\left|f^{(k)}-\stackrel{{\scriptstyle i}}_{nn}^{(k)}\right|\leqq(n+1-k)M_{\omega_{k}}(nh),\text{ pentru }x\in\left[\lambda_{j+n},\lambda_{j+n+1}\right]

j=j_{0},j_{0}-1-1,\ldots,j_{1}

so, all the more so,

\left|f^{(k)}-\psi_{m}^{(k)}\right|\leqq(n+1-k)M\omega_{k}(nh),\text{ pentru }x\in\left[a^{\prime},b^{\prime}\right]

(77)

which, based on the well-known properties of the oscillation modulus, of continuous functions, proves the theorem in this case.

It is easy to see that the delimitation (77) is also valid in the other possible cases. The modifications that need to be made to the proof are the following:

If $b^{\prime}=b$ , the term is deleted $\frac{b-a}{b^{\prime\prime}-b^{\prime}}$ , in the second member of formula (76).

If $a^{\prime}=a$ , the term is deleted $\frac{n(b-a)}{a^{\prime}-a^{\prime\prime}}$ in the second member of formula (76) and it is observed that for $j<0$ NUMBER $r$ is subject to the condition that $0\leqq r\leqq n-k$ . Divided difference nodes $D_{k}^{r}[/]$ I am then in the range $\left[\bar{\lambda}_{0},\lambda_{n}\right]$ .

Theorem 14 is proved.
30. We can now return to the study of the simplicity criteria of linear functionals.

Whether $[a,b]$ a finite and closed interval and consider the non-ascending sequence of $k+1$ partial intervals $\left[a_{0},b_{0}\right]\supseteqq\left[a_{1},b_{1}\right]\supseteq\ldots\supseteq\left[a_{k},b_{k}\right]$ , where $a_{0}=a,b_{0}=b$ .

Whether $\bigodot_{k}$ function space $/$ which admit continuous derivatives of the order $i$ on $\left[a_{i},b_{i}\right]$ for $i=0,1,\ldots,k$ and let's consider the norm

\|f\|=\sum_{i=0}^{k}\max_{x\in\left[a_{i},b_{i}\right]}\left|f^{(i)}\right|

(78)

of this space.

we

THEOREM 15. - Find the given natural number $n$ and the whole $k$ so that $0\leqq h\leqq n-1$ , if the linear functional $R[f]$ is: $1^{\circ}$ . defined on $@_{k}$ , $2^{\circ}$ . degree of accuracy $n,3^{\circ}$ . bounded with respect to the norm (78) because $R[f]$ to be of simple form is necessary and sufficient for us to have

R\left[x^{n+1}\right]R\left[\varphi_{n+1,\lambda}\right]\geqq 0,\text{ pentru }\lambda\in[a,b],

(79)

where the functions $\varphi_{n+1,n}$ are defined by formula (64).
Let us note that the polynomials and functions $\varphi_{n+1,\lambda}$ , belong to the space $e_{k}$ .
The condition is necessary. Indeed, $x^{n+1}$ is convex and $\varphi_{n+1,\lambda}$ is non-concave of the order $n$ The property results from formula (31).

The condition is also sufficient. By hypothesis, we have

|R[f]|\leqq A\|f\|,\quad t\in\mathscr{C}_{k},

$A$ being a number independent of the function $f$ and $\|f\|$ norma (78).
Vom demonstra întîi că $R\left[\varphi_{n+1,\lambda}\right]$ este o funcție continuă de $\lambda$ pe $[a,b]$ . Intr-adevăr, avem

\varphi_{n+1,\lambda}-\varphi_{n+1,\lambda^{\prime}}|\leqq n|\lambda-\lambda^{\prime}\mid(b-a)^{n-1},x\in[a,b]

deci și $(0\leqq i\leqq n-1)$ ,

		$\displaystyle\left\|\varphi_{n+1,\lambda}^{(i)}-\varphi_{n+1,\lambda^{\prime}}^{(i)}\right\|=\frac{n!}{(n-i)!}\left\|\varphi_{n+1-i,\lambda}-\varphi_{n+1-i,\lambda^{\prime}}\right\|\leqq$
		$\displaystyle\leq\frac{n!}{(n-i-1)!}\left\|\lambda-\lambda^{\prime}\right\|(b-a)^{n-1-i},\quad x\in[a,b].$

Avem deci
$\left|R\left[\varphi_{n+1,\lambda}\right]-R\left[\varphi_{n+1,\lambda^{\prime}}\right]\right|=\left|R\left[\varphi_{n+1,\lambda}-\varphi_{n+1,\lambda^{\prime}}\right]\right|\leqq A\left\|\varphi_{n+1,\lambda}-\varphi_{n+1,\lambda^{\prime}}\right\|$ .
Dar,

\left\|\varphi_{n+1,\lambda}-\varphi_{n+1,\lambda^{\prime}}\right\|\leqq\left[\sum_{i=0}^{k}\frac{n!}{(n-i-1)!}(b-a)^{n-1-i}\right]\left|\lambda-\lambda^{\prime}\right|

de unde proprietatea rezultă fără nici o dificultate.
Prin ipoteză, $R\left[x^{n+1}\right]\neq 0$ , deci $R\left[\varphi_{n+1,\lambda}\right]$ nut schimbă de semn cînd $\lambda$ parcurge intervalul $[a,b]$ . Să ne reamintim că o funcție convexă de ordinul $n$ pe $[a,b]$ are o derivată continuă de toate ordinele $\leq n-1$ pe $(a,b)$ . Dacă deci $f\in e_{k}$ este convex de ordinul $n$ , în virtutea teoremei 14 şirul $\left\{R\left[\psi_{m}\right]\right\}$ tinde către $R[f]$ pentru $m\rightarrow\infty$ . Dar, pe baza formulelor (68) - (70), avem $R\left[\psi_{m}\right]=R[f]=(n+1)h^{m-n-1}D_{n+1}^{i}[f]R\left[\varphi_{n+1,\lambda_{i+n}}\right]$ şi din (79) rezultă că dacă $f$ este convex de ordinul $n$ , avem

R\left[x^{n+1}\right]R[f]\geqq 0.

(80)

Rămîne să demonstrăm că în această formulă egalitatea nu poate avea loc. Am dat această demonstraţie în altă parte [15], aşa că mu mai revenim aici asupra ei.

Se deduce că pentru orice funcție $f\in\mathbb{C}_{k}$ convexă de ordinul $n$ semmul $>$ este valabil în (80), deci că $R[/]\neq 0$ .

Yeorema 15 este deci demonstrată.
31. Fie $R[/]$ o funcţională liniară definită pe $e_{k}$ şi mărginită in raport cu norma (78). Să presupunem că $0\leqq k\leqq n-1$ şi că $a=a_{0}==a_{1}=\ldots=a_{k},b=b_{0}=b_{1}=\ldots=b_{k}$ . Atunci, după E. Ya. Remez [22], dacă $R[f]$ este de gradul de exactitate $n$ , avem

R[f]=\int_{a}^{b}f^{(\mu)}(x)d\alpha_{\mu}(x)

(81)

unde $\mu$ este un întreg, $k\leqq\mu\leqq n+1$ si $\alpha_{\mu}$ o functie cu variaţia mărginită care, pentru $\mu\leqq\lambda$ , verifică egalitatea $\alpha_{\mu}(a)=\alpha_{\mu}(b)=0$ . Reprezentarea (81) este valabilă dacă derivata a $\mu-a,f^{(\mu)}$ este continuă pe $[a,b]$ . E. Ya. Remez a demonstrat [22] şi formulele

	$\displaystyle\alpha_{\mu+1}(x)=-\int_{a}^{b}\alpha_{\mu}(x)dx,\quad k\leqq\mu\leqq n$		(82)
	$\displaystyle R[f]=-\int_{a}^{b}f^{(\mu)}(x)\alpha_{\mu-1}(x)dx,\quad k+1\leqq\mu\leqq n+1$		(83)

In particular, funcția de $\lambda\varphi_{n+1,\lambda}$ admite o derivată continuă de ordinul $n-1$ pe $[a,b]$ . Avem deci, ținînd cont de (80), (81),

	$\displaystyle R\left[\varphi_{n+1,\lambda}\right]=$	$\displaystyle n!\int_{\lambda}^{b}(x-\lambda)d\alpha_{n-1}(x)=-n!\int_{\lambda}^{b}\alpha_{n-1}(x)dx=$
		$\displaystyle=n!\int_{a}^{\lambda}\alpha_{n-1}(x)dx=-n!\alpha_{n}(\lambda)$

Din (83) rezultă deci că dacă $f$ are o derivată de ordinul $n+1$ continux pe $[a,b]$ , avem reprezentarea

R[f]=\frac{1}{n!}\int_{a}^{b}R\left[\varphi_{n+1,\lambda}\right]f^{(n+1)}(x)dx

(84)

32.

Să reluăm formula (56) a lui Gauss. Am stabilit formula (57) sub ipoteza continuitătii funcţiei $f$ pe $[-1,1]$ si a derivatei sale pe $(-1,1)$ . Insă în cazul acesta functionala liniară $R[f]$ este mărginită pe spatiul $e_{0}$ al functiilor continue pe $[-1,1]$ , in raport cu norma max $|f|$ .

Formula (57) este, în particular, adevărată pentru funcțiile $f=\varphi_{2m,\lambda}$ care sînt neconcave de ordinul $2m-1$ . Se deduce că $R\left[\varphi_{2m,\lambda}\right]\geqq 0$ pentru $\lambda\in[-1,1]$ și, aplicînd teorema 15, rezultă că formula (57) este adevărată sub singura ipoteză a continuitătii functiei f pe intervalul $[a,b]$ .

§4.

83.

Vom examina în acest §, fără a întra în prea multe detalii, cazul cînd funcţionala liniară $R[f]$ nu este de forma simplă.

O funcţională liniară $R_{1}[f]$ definită pe $\mathscr{F}$ se numește o majorantă simplă a lui $R[f]$ dacă : $1^{\circ}$ . ea este de forma simplă, $2^{\circ}$ . avem $R_{1}[f]>R[f]$ pentru orice funcţie convexă $f\in\sqrt{f}$

Avem atunci

TEOREMA 16. - Dacă functionala liniară $R[f]$ definită pe $f$ admite majorantă simplă, avem

R[f]=K\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};f\right]-K^{\prime}\left[\xi_{1}^{\prime},\xi_{2}^{\prime},\ldots,\xi_{n+2}^{\prime};f\right]

(85)

unde : $1^{\circ}.K,K^{\prime}$ sint numere diferite de zero și independente de functia $f,2^{\circ}$ . punctele $\xi_{i}\in E,i=1,2,\ldots n+2$ pe de o parte și punctele $\xi_{i}\in E,i==1,2,\ldots,n+2$ pe altă parte, sînt distincte (ele pot depinde, în general, de functia $f$ ).

Intr-adevăr, fie $R_{1}[f]$ o majorantă simplă a lui $R[f]$ . Avem $R[f]==R_{1}[f]-\left\{R_{1}[f]-R[f]\right\}$ , unde functionalele liniare $R_{1}[f],R_{1}[f]-$ - $R[f]$ sînt de forma simplă.

Să considerăm o funcțională liniară $R[f]$ definită pe $\mathbb{F}$ şi de forma (85) indicată în teorema 16. Dacă constantele $K,K^{\prime}$ sînt de semne contrare, $R[f]$ este de forma simplă. Este deci destul să examinăm cazul cînd $K,K^{\prime}$ sînt (diferite de zero şi) de acelaşi semn. Fără să restrîngem generalitatea, putem atunci presupune că ei sînt pozitivi. Avem atunci

L e m a 4. - Dacă functionala liniară $R[f]$ is defined on the space (f and if it is of the form (85), indicated in Theorem 16, for any function / $\in$ with the bounded divided difference,
the representation (85) is valid for any $t\in\mathscr{F}$ (so also for the elements $1ui$ $\widehat{f}$ which do not have bounded divided difference).

It is easy to see that Lemma 4 is a consequence of the following:
Lemma 5. - If: $1^{\circ}$ . R[f] is a linear functional defined on ${f}$ , $2^{\circ}.K,K^{\prime}$ are two positive numbers,
for any $f\in\mathbb{F}$ whose divided difference is not bounded, one can find $n+2$ distinct points $\xi_{i}\in E,i=1,2,\ldots,n+2$ and $n+2$ distinct points $\xi_{i}^{\prime}\in E,i=1,2,\ldots,n+2$ , so that we have (85).

Let us assume, for the sake of clarity, that the divided difference of the function $f$ is not bounded above, By virtue of Theorem 4, if the divided difference of this function takes the value $m$ , it will take any value greater than $m$ . Be it then $m$ a value taken from the difference

12 - Mathematics studies and research
$\left[\xi_{1}^{\prime},\xi_{2}^{\prime},\ldots,\xi_{n+2}^{\prime};f\right]$ a divided difference that takes a value $>\frac{mK-R[f]}{K^{\prime}}$ and $\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};t\right]$ a divided difference that takes the value

\frac{1}{K}\left\{K^{\prime}\left[\xi_{1}^{\prime},\xi_{2}^{\prime},\ldots,\xi_{n+2}^{\prime};f\right]+R[f]\right\}>m.

Formula (85) results.
The same procedure is followed if the divided difference of the function $f$ it is not bounded inferiorly.

Lemma 5 is therefore proven.
We recall that the notion of divided difference, of simplicity of linear functionals and the spaces considered are in the sense of § 1.

It is clear that instead of simple majorities we can use simple minorities. Linear functional $R_{1}[f]$ defined on $\mathscr{F}$ is called a simple minor for $R[t]$ if it is of simple form and if $R_{1}[t]<<R[f]$ for any concave function $f$ .

To be able to put a linear functional $R[f]$ in the form ( 85 ), it is therefore sufficient to know a simple majorant (or a minorant). For example, the linear functional (58), which cancels out on the functions (19) and which can therefore be put in the form (59), has as a simple majorant the linear functional

\sum_{i=1}^{m-n-1}\frac{\left|\mu_{i}\right|+\mu_{i}}{2}\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]+\mu\left[x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+2}^{\prime};f\right],

where $\mu$ is a positive number and $x_{i}^{\prime},n+2$ distinct points of the interval $E$ . All linear functionals of the form (58) can therefore be put into the form (85), indicated in Theorem 16.
34. If the linear functional $R[f]$ is of the form (85), the difference $K-K^{\prime}=R\left[f_{n+1}\right]$ has a perfectly determined value. Suppose that $K,K^{\prime}$ are positive. We can then replace $K,K^{\prime}$ by $K+\varepsilon,K^{\prime}+\varepsilon$ respectively, $\varepsilon$ being an arbitrary positive number. Indeed, if we have (85) for a $f\in\mathbb{F}$ given, we have $R[f]=R_{1}[f]-R_{2}[f]$ , where $R_{1}[f]==K\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};f\right]+\varepsilon\left[x_{1},x_{2},\ldots,x_{n+2};f\right],R_{2}[f]=K^{\prime}\left[\xi_{1}^{\prime},\xi_{2}^{\prime},\ldots,\xi_{n+2}^{\prime};f\right]++\varepsilon\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$ , where $x_{i}$ saint $n+2$ distinct points of the interval $E$ We can look at $R_{1}[f],R_{2}[f]$ as linear functionals defined on $\mathscr{F}$ . Then they are of the simple form. The stated property results from observing that $R_{1}\left[f_{n+1}\right]=K+\varepsilon,R_{2}\left[f_{n+1}\right]=K^{\prime}+\varepsilon$ .

If $R\|f]$ is of the form (85) but is not of simple form, the coefficients $K,K^{\prime}$ , assumed positive, have some lower bounds whose values are of interest especially when $R[f]$ is the remainder of an approximation formula. In this sense we will examine an important particular case in the next no.
35. Let us suppose again that we are in the particular case (21), (21') and consider a linear functional $R[f]$ defined and bounded in space $e_{k}$ considered at No. 30. We have

THEOREM 17. - If: $1^{\circ}$ . linear functional $R[f]$ is defined on $\bigodot_{k}$ , limited in relation to the norm (78) and the degree of accuracy $n$ , with $R\left[x^{n+1}\right]>0,2^{\circ}$ . A is the upper edge of $R[f]$ for the functions $f\in\mathbb{C}_{k}$ whose difference divided by the order $n+1$ remains contained in $[0,1]$ and $B=A-R\left[x^{n+1}\right]$ ,
for anything $\varepsilon>0$ , linear functional $R[f]$ is of the form (85), indicated by Theorem 16, where $K=A+\varepsilon,K^{\prime}=B+\varepsilon$ .

From the demonstration it will follow that $A,B$ are finite.
We have $A>0$ because, in particular, $x^{n+1}$ has its difference divided by the order $n+1$ contained in $[0,1]$ We have obviously $B\geq 0$ .

If we consider the functions (69), by the formula $R_{m}[f]=R\left[f_{m}\right]$ we define a linear functional which, for $m\rightarrow\infty$ , tends towards $R[f]$ for anything $f\in e_{k}$ Let's put

R_{m}^{+}[f]=(n+1)h\cdot\sum_{i=1}^{m-1}D_{n+1}^{i}[f]\frac{R\left[\varphi_{n+1,\lambda_{i+n}}\right]+\left|R\left[\varphi_{n+1,\lambda_{i+n}}\right]\right|}{2}

(86)

and let's note with $e_{k}^{*}$ his/her subset $e_{k}$ made up of the functions $f$ which have their differences divided by the order $n+1$ bounded. Moreover, any function defined on $[a,b]$ , having its difference divided by the order $n+1$ bordered, belongs to $\ell_{k}$ Let us note that the function $x,R\left[\varphi_{n+1,x}\right]$ being continuous on $[a,b]$ , the string with positive terms

\left\{(n+1)h\sum_{i=1}^{m-n-1}\frac{R\left[\varphi_{n+1;\lambda_{i+n}}\right]+\left|R\left[\varphi_{n+1,\lambda_{i\div n}}\right]\right|}{2}\right\}

(87)

tend, for $m\rightarrow\infty$ , towards a finite and well-determined limit equal to

A=(n+1)\int_{a}^{b}\frac{\left.R\left[\varphi_{n+1,}x\right]+|R|\varphi_{n+1},x\right]\mid}{2}dx

(88)

It follows that the sequence (87) is bounded. If $f\in\mathcal{C}_{k}^{*}$ , the string $\left\{R_{m}^{+}[f]\right\}$ is also bounded. One can extract from this sequence a partial sequence convergent to the functional $R^{+}[f]$ It is easy to see that the functional $R^{+}[f]$ thus defined on $e_{k}^{*}$ is linear and vanishes on any polynomial of degree $n$ But we have $R_{m}^{+}[f]>0,R_{m}^{\star}[f]\geqq R_{m}[f]$ if $f\in e_{k}$ is convex, so $R^{+}[f]\geqq 0,R^{+}[f]\geqq R[f]$ if $f\in\mathbb{C}_{k}^{*}$ is convex. It immediately follows that if $\varepsilon$ is a positive number and $x_{1},x_{2},\ldots,x_{n+2},n+2$ fixed points of the interval $E$ , the linear functional $R_{1}[f]=R^{+}[f]+\varepsilon\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$ is a simple majority of $R[f]$ It is easy to see that $R_{1}\left[x^{n+1}\right]=A+\varepsilon$ , where $A$ is given by formula (88).

It remains to be proven that the number $A$ , given by formula (88), coincides with the upper edge of $R[f]$ if $f$ traverses the set of functions whose difference divided by the order $n+1$ remains contained in $[0,1]$ If $f$ is such a function, it is clear that $R_{m}[f]$ does not exceed the general (corresponding) term of the sequence ( 87 ). Taking the limit, it follows that $R[f]$ don't blink -
step on $A$ . Be it now $\varepsilon$ an arbitrary positive number. Let us take into account the continuity of the function $x,R\left[\varphi_{n+1,x}\right]$ , therefore by the continuity and non-negativity of the function $x,\frac{1}{2}\left\{R\left[\varphi_{n+1,x}\right]+\mid R\left[\varphi_{n+1,x}\right]\right\}$ and note that the points at which a function continues on $[a,b]$ cancels out, forms $o$ closed set. It follows that we can find a finite number $k$ of disjoint intervals $\left[\alpha_{i},\beta_{i}\right],i=1,2,\ldots,k$ , belonging to $(a,b)$ and so that the function $R\left[\varphi_{n+1,x}\right]$ to be nonnegative on these intervals and so that we have

(n+1)\sum_{i=1}^{k}\int_{a_{i}}^{\beta_{i}}R\left[\varphi_{n+1,x}\right]dx>A-\frac{\varepsilon}{2}

(89)

We can assume $a<\alpha_{1}<\beta_{1}<\alpha_{2}<\beta_{2}<\ldots<\alpha_{k}<\beta_{k}<b$ . Either $M=\max_{(n+1)\left|R\left[\varphi_{n+1,x}\right]\right|\text{ si să alegem punctele }\alpha_{i}^{\prime},\quad\beta_{i}^{\prime}\text{, }}i=1,2,\ldots,k$ so that we have $a<\alpha_{1}^{\prime}<\alpha_{1},\beta_{k}<\beta_{k}^{\prime}<b,\beta_{i-1}<\beta_{i-1}^{\prime}<<\alpha_{i}^{\prime}<\alpha_{i},\quad i=2,3,\ldots,k$ and as

M\sum_{i=1}^{k}\left(\alpha_{i}-\alpha_{i}^{\prime}+\beta_{i}^{\prime}-\beta_{i}\right)<\frac{\varepsilon}{2}.

(90)

Be it now $f$ a function whose derivative of the order $n+1$ exists and is continuous on $[a,b]$ , this derivative reducing to $(n+1)$ ! on the intervals $\left[\alpha_{i},\beta_{i}\right],i=1,2,\ldots,k$ , to 0 on the intervals $\left[a,\alpha_{j}^{\prime}\right],\left[\beta_{k}^{\prime},b\right]$ , $\left[\beta_{i-1}^{\prime},\alpha_{i}\right],i=2,3,\ldots,k$ and one linear function on each of the intervals $\left[\alpha_{i}^{\prime},\alpha_{i}\right],\left[\beta_{i},\beta_{i}^{\prime}\right],i=1,2,\ldots,k$ Function $f$ considered to belong to $e_{k}$ and the formula (29) of the average shows us that the difference divided by the order $n+1$ remains contained in $[0,1]$ Taking into account the representation ( 85 ), we have for this function

	$\displaystyle R[f]=(n+1)\sum_{i=1}^{k}\int_{a_{i}}^{\beta_{i}}R\left[\varphi_{n+1,x}\right]dx+$
	$\displaystyle+(n+1)\left\{\sum_{i=1}^{k}\int_{a_{i}^{\prime}}^{a_{i}}R\left[\varphi_{n+1,x}\right]\frac{f^{(n+1)}(x)}{(n+1)!}dx+\sum_{i=1}^{k}\int_{\beta_{i}}^{\beta_{i}^{\prime}}R\left[\varphi_{n+1,x}\right]\frac{f^{(n+1)}(x)}{(n+1)!}dx\right\}.$		(91)

But

\begin{gathered}(n+1)\left|\sum_{i=1}^{k}\int_{a_{i}}^{a_{i}}+\sum_{i=1}^{k}\int_{\beta_{i}}^{\beta_{i}^{\prime}}R\left[\varphi_{n+1,x}\right]\frac{f^{(n+1)}(x)}{(n+1)!}dx\right|\leqq\\ \leqq(n+1)\left\{\sum_{i=1}^{k}\int_{a_{i}^{\prime}}^{a_{i}}+\sum_{i=1}^{k}\int_{\beta_{i}}^{\beta^{\prime}}\left|R\left[\varphi_{n+1,x}\right]\right|dx\right\}\leqq M\sum_{i=1}^{k}\left(\alpha^{i}-\alpha_{i}^{\prime}+\beta_{i}^{\prime}-\beta_{i}\right)<\frac{\varepsilon}{2}\end{gathered}

Taking into account (89), (92), from formula (91) it follows that $R[f]>A-\varepsilon$ . Number $A$ is therefore the upper bound indicated in the statement of the theorem.

Theorem 17 is therefore proven.
In this theorem we assumed $R\left[x^{n+1}\right]>0$ . Otherwise, so if $R\left[x^{n+1}\right]<0$ , the property and the proof are analogous. In this case $B>0,A\geqq 0$ .

In cases $A=\overline{0}$ or $B=0$ , functional $R[f]$ is of simple form.
It is easy to show that if $f\in C_{k}$ has a derivative of order $n+1$ continue on $[a,b]$ , we have

R[f]=\frac{1}{(n+1)!}\left\{Af^{(n+1)}(\xi)-Bf^{(n+1)}(\eta)\right\},\quad\xi,\eta\in[a,b]

If $f\epsilon e_{k}^{*}$ and if $d$ is the upper bound of the absolute value of the difference divided by the order $n+1$ his/her $t$ , we have the delimitation

|R[f]|\leqq(A+B)d

36.

There are other forms in which a linear functional can be put $R[f]$ , so the remainder of a linear approximation formula. These expressions are of interest especially when $R[f]$ it is not of simple form.

Let us suppose that we are in the particular case (21), (21') and let us suppose that $R[f]$ is a linear functional defined and of degree of accuracy $n$ on $\widehat{f}$ Let us consider a decomposition of the form

R[f]=R_{1}[f]+\left\{R[f]-R_{1}[f]\right\}

(93)

where $R_{1}[t]$ is a linear functional defined on $(\tau$ and where the linear functional (also defined on $(f)R[f]-R_{1}[f]$ has a degree of accuracy $n+p>n$ . Then if $R_{1}[f]$ and $R[f]-R_{1}[f]$ are of simple form, we have

R[f]=R\left[x^{n+1}\right]\left[\xi_{1},\xi_{2},\ldots,\xi_{n+2};f\right]+K\left[\xi_{1}^{\prime},\xi_{2}^{\prime},\ldots,\xi_{n+p+2}^{\prime};f\right]

(94)

where $K\neq 0$ is independent of the function $f$ and $\xi_{i},\xi_{i}$ are groups of $n+2$ resp. $n+p+2$ distinct points in $E$ .

Without claiming to make a general theory here, we will show, by two examples, how a representation of the form (94) can actually be found for the rest of certain approximation formulas.
37. Let us consider Hardy's quadrature formula,

\int_{0}^{6}f(x)dx=0,28[f(0)+f(6)]+1,62[f(1)+f(5)]+2,2f(3)+R[f]

The degree of accuracy of the remainder $R[f]$ is 5. A simple calculation shows us that $R\left[\varphi_{6,3}\right]=\frac{81}{50}>0,\quad R\left[\varphi_{6,5}\right]=-\frac{17}{150}<0$ and so, by virtue of Theorem 15, the remainder is not of simple form.

To put $R[f]$ in the form (94) it is advantageous to first consider the linear functional $R^{*}[f]=R\left[f^{\prime}\right]$ , which we have already considered in the previous §. Indeed, it is enough to find a decomposition of the form (94) for this linear functional. The corresponding decomposition for $R[f]$ , it results immediately.

we
$R^{*}[f]=-63[0,0,1,1,3,3,5,5;f]+190,8[0,1,1,3,3,5,5,6;f]-$

-63[1,1,3,3,5,5,6,6;f]

Whether

	$\displaystyle R_{1}[f]=\mu_{1}[0,0,1,1,3,3,5,5;f]+\mu_{2}[0,1,1,3,3,5,5,6;f]+$
	$\displaystyle+\mu_{3}[1,1,3,3,5,5,6,6;f]$		(95)

where
we have then $\mu_{1}+\mu_{2}+\mu_{3}=64,8$ .

R^{*}[f]-R_{1}[f]=6\left(63+\mu_{1}\right)[0,0,1,1,3,3,5,5,6;f]-

(96)

-6\left(63+\mu_{3}\right)[0,1,1,3,3,5,5,6,6;f]

(97)

which has a degree of accuracy $>6$ The linear functionals
(95), (97) are of simple form if $\mu_{1}=\mu_{3},\mu_{2}$ are nonnegative. We thus find the following expression for the remainder in Hardy's formula,

R[f]=\frac{9}{700}\left\{6!\left[\xi_{1},\xi_{2},\ldots,\xi_{2};f\right]-\frac{5\left(63+\mu_{1}\right)}{648}8!\left[\eta_{1},\eta_{2},\ldots,\eta_{9};f\right]\right\}

where $f$ is continuous on $[0,6],\xi_{i}$ there are 7 distinct points and $\eta_{i}$ There are 9 distinct points in the interval (0.6).

From the particular method of demonstration it follows that in this formula we have $0\leqq\mu_{1}\leqq 32,4$ .

If $f$ has a continuous derivative of order 8 on ( 0.6 ), we have

R[f]=\frac{9}{700}\left\{f^{(6)}(\xi)-\frac{5\left(63+\mu_{1}\right)}{648}f^{(8)}(\eta)\right\},\quad\xi,\eta\in(0,6)

If we put in this formula $\mu_{1}=\frac{9}{5}$ , we find the well-known remainder [24]

R[f]=\frac{9}{700}\left\{f^{(6)}(\xi)-\frac{1}{2}f^{(8)}(\eta)\right\},\quad\xi,\eta\in(0,6)

(98)

But we can also take $\mu_{1}=0$ and then we find

R[f]=\frac{9}{700}\left\{f^{(6)}(\xi)-\frac{35}{72}f^{(8)}(\eta)\right\},\quad\xi,\eta\in(0,6)

where the coefficient $\frac{35}{72}$ is smaller than the corresponding coefficient $\frac{1}{2}$ from formula (98).
38. As a second application, let us take Weddle's quadrature formula,
⁶
$\int_{0}^{6}f(x)dx=0,3[f(0)+f(2)+f(4)+f(6)]+1,5[f(1)+f(5)]+1,8f(3)+R[f]$ , whose remainder is still of accuracy degree 5. We have $R\left[\varphi_{6,3}\right]=\frac{3}{10}>0$ , $R\left[\varphi_{6,4}\right]=-\frac{13}{30}<0$ , so the remainder is not of simple form. Proceeding as in the previous example, we have

\begin{gathered}\frac{5}{18}R^{*}[t]=-3[0,0,1,1,2,2,3,3;t]-4[0,1,1,2,2,3,3,4;t]+\\ +4[1,2,2,3,3,4,4,5;t]-4[2,3,3,4,4,5,5,6;t]-\\ -3[3,3,4,4,5,5,6,6;t]\end{gathered}

and we take

	$\displaystyle\frac{5}{18}R_{1}[f]=\mu_{1}[0,0,1,1,2,2,3,3;f]+\mu_{2}[0,1,1,2,2,3,3,4;f]+$
	$\displaystyle+\mu_{3}[1,1,2,2,3,3,4,4;f]+\mu_{4}[1,2,2,3,3,4,4,5;f]+$		(99)
	$\displaystyle+\mu_{3}[2,2,3,3,4,4,5,5;f]+\mu_{2}[2,3,3,4,4,5,5,6;f]+$
	$\displaystyle+\mu_{1}[3,3,4,4,5,5,6,6;f]$
	$\displaystyle 2\left(\mu_{1}+\mu_{2}+\mu_{3}\right)+\mu_{4}=-10$

and then we have

$\displaystyle-\frac{5}{18}\left[R^{*}[f]\right.$	$\displaystyle\left.-R_{1}[f]\right]=16\left(\mu_{1}+3\right)[0,0,1,1,2,2,3,3,4,4;f]+$	(101)
	$\displaystyle+20\left(2\mu_{1}+\mu_{2}+10\right)[0,1,1,2,2,3,3,4,4,5;f]+$
	$\displaystyle+16\left(3\mu_{1}+2\mu_{2}+\mu_{3}+17\right)[1,1,2,2,3,3,4,4,5,5;f]+$
	$\displaystyle+20\left(2\mu_{1}+\mu_{2}+10\right)[1,2,2,3,3,4,4,5,5,6;f]+$
	$\displaystyle+16\left(\mu_{1}+3\right)[2,2,3,3,4,4,5,5,6,6;f]$

Let's take $\mu_{1}=-3,\mu_{2}=-2,\mu_{3}=\mu_{4}=0$ Then (100) is true.

R[f]=-\frac{1}{140}\left\{6!\left[\xi_{1},\xi_{2},\ldots,\xi_{7};f\right]+\frac{1}{5}8!\left[\eta_{1},\eta_{2},\ldots,\mu_{3};f\right]\right\}

function $/$ and the points $\xi_{i},\eta_{i}$ verifying the same conditions as in the previous example (no. 37). If the function $f$ has a continuous derivative of order 8 on $(0,6)$ , we have

R[f]=-\frac{1}{140}\left\{f^{(6)}(\xi)+\frac{1}{5}f^{(8)}(\eta)\right\},\quad\xi,\eta\in(0,6)

In the well-known formula [24],

R[t]=-\frac{1}{140}\left\{f^{(6)}(\xi)+\frac{9}{10}f^{(8)}(\eta)\right\}\quad\xi,\eta\in(0,6)

the coefficient of the 8th order derivative is 4.5 times greater in absolute value.

Let us also observe that if, in addition to (100), we also have

20\mu_{1}+9\mu_{2}+2\mu_{3}=-96,

(102)

we can write

	$\displaystyle-\frac{5}{18}\left\{R^{*}[f]-R_{1}[f]\right\}=400\left(\mu_{1}+3\right)[0,0,1,1,2,2,3,3,4,4,5,5;f]+$
	$\displaystyle+120\left(18\mu_{1}+5\mu_{2}+74\right)[0,1,1,2,2,3,3,4,4,5,5,6;f]+$		(103)
	$\displaystyle+400\left(\mu_{1}+3\right)[1,1,2,2,3,3,4,4,5,5,6,6;f]$

If we take $\mu_{1}=-\frac{51}{11},\mu_{2}=-\frac{4}{11},\mu_{3}=\mu_{4}=0$ , the equalities (100), (102) are verified and the linear functionals (99), (103) are of simple form. For the rest $R[t]$ of Weddle's formula we obtain

R[f]=-\frac{1}{140}\left\{6!\left[\xi_{1},\xi_{2},\ldots,\xi_{7};f\right]-\frac{61}{1815}10!\left[\eta_{1},\eta_{2},\ldots,\eta_{11};f\right]\right\}

where $f$ is continuous on $[0,6],\xi_{i}$ there are 7 distinct points and $\eta_{i}$ there are 11 distinct points of the interval $(0,6)$

If the function $f$ has a continuous derivative of order 10 on ( 0.6 ), we have

R[f]=-\left\{\frac{1}{140}f^{(6)}(\xi)-\frac{61}{8115}f^{(10)}(\eta)\right\},\quad\xi,\eta\in(0,6).

On the remainder in some linear approximation formulas of the analysis

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ABOUT THE REST IN SOME LINEAR APPROXIMATION FORMULAS OF ANALYSIS

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