ON A GENERALIZATION OF GAUSS'S NUMERICAL INTEGRATION FORMULA

BY
TIBERIU POPOVICIU
Communication presented on May 4, 1955 at the meeting of the 1955 Branch of the Academy of the Romanian Republic

§ 1. Gaussian-type formulas.

1.

In practical numerical calculation problems, the value of a linear functional is often required ¹ ) $\mathrm{A}[f]$ , defined on a vector space $S$ of functions $f=f(x)$ , real, of the real variable $x$ , defined and continuous on an interval I. In what follows we will assume that the elements of $S$ are differentiable a sufficient number of times, at least at the points where these derivatives will occur. We will also assume that $S$ contains all polynomials in $x$ In the following we will impose the functionality $\vec{A}[f]$ and certain restrictive conditions, which we will specify when they occur.
2.

Suppose that the values are given

f^{(j)}\left(x_{i}\right),j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n\quad\left(f^{(0)}(x)=f(x)\right),

(1)

of the function $f(x)$ and its first derivatives $f^{\prime}(x),f^{\prime\prime}(x),\ldots$ on distinct points
(2)

x_{1},x_{2},\ldots,x_{n}

of interval I. On the point $x_{i}$ the values of the function and its primes are given $r_{i}-1$ derived, so that the numbers $r_{i},i=1,2,\ldots,n$ are entirely positive.

For each $f\in S,\bar{A}[f]$ can be approximated by a given linear combination of the values (1) of the function $f(x)$ and its first derivatives on points (2). We thus obtain the approximation formula

\mathrm{A}[f]\approx\sum_{i=1}^{n}\sum_{j=0}^{r_{i}-1}a_{i,j}f^{(j)}\left(x_{i}\right).

(3)

The points (2) are the nodes of this formula and the numbers $r_{i},i=1,2,\ldots,n$ are the multiplicity orders of these nodes. The node $r_{i}$ has the order to

⁰⁰footnotetext: 1. By a linear functional we mean an additive and homogenic functional.

multiplicity $r_{i},i=1,2,\ldots,n$ . The numbers $a_{i,j},j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n$ characterizes the approximation formula (3) of this type and can be called the coefficients of this formula.

REST $\mathrm{R}[f]$ of formula (3) is, by definition, the difference between the first and second members of the formula. If we then add the remainder to the second member of the formula, this approximate equality becomes an ordinary equality. The remainder is, obviously, also a linear functional defined on $S$ .
3. Another interpretation of the approximation formula (3) can be given. A general method for finding an approximate value for $\mathrm{A}[f]$ consists of replacing the function $f(x)$ through another function $\varphi\in S$ and take the $\mathrm{A}[\varphi]$ as an approximate value for $\mathrm{A}[f]$ , so $\mathrm{A}[f]\approx\mathrm{A}[\varphi]$ .

From a theoretical point of view, nothing prevents us from choosing the function $\varphi(x)$ completely arbitrary, with the only restriction that it belongs to $S$ . But from the point of view of practical applications, the choice of functions $\varphi(x)$ must and can generally be considerably restricted. An important case is when $\varphi(x)=\mathrm{B}[f\mid x]$ is a given linear operator, with values in $S$ . In this case $\mathrm{A}[\varphi]=\mathrm{A}[\mathrm{B}[f\mid x]]$ is a linear functional of $f$ , well determined and defined on $S$ The rest $\mathrm{R}[f]=\mathrm{A}[f]-\mathrm{A}[\varphi]$ will then also be a well-determined and defined linear functional on $S$ .

An important class of functions $\varphi(x)$ of the previous form is formed by the generalized linear interpolation functions

\varphi(x)=\mathrm{B}[f\mid x]=\sum_{i=1}^{n}\sum_{j=0}^{ri-1}\varphi_{i,j}(x)f^{(j)}\left(x_{i}\right)

(4)

corresponding to the nodes (2), with the indicated multiplicity orders and where $\varphi_{i,j}(x)\in S,j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n$ are some given functions. Then the formula $A[f]\approx A[\varphi]$ reduces to (3), where the coefficients are given by the formulas

a_{i,j}=\mathrm{A}\left[\varphi_{i},j\right],j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n.

In particular, either

\varphi(x)=\mathrm{L}(\underbrace{x_{1},x_{1},\ldots,x_{1}}_{r_{1}},\underbrace{x_{2},x_{2},\ldots,x_{2}}_{r_{2}},\ldots,\underbrace{x_{n},x_{n},\ldots,x_{n}}_{r_{n}};f\mid x)

the Lagrange-Hermite interpolation polynomial of degree

p=r_{1}+r_{2}+\cdots+r_{n}-1,

(5)

which, together with its first $r_{i}-1$ derivatives, the respective values (1) on the nodes $x_{i},i=1,2,\ldots,n$ This function $\varphi(x)$ is of the form (4), where

\varphi_{i,j}(x)==l_{i,j}(x),j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n

are the fundamental interpolation polynomials relative to the nodes (2) with their respective multiplicity orders. These polynomials are completely determined. They have some well-known expressions, among which we recall the formulas
(6) $\quad l_{i},r_{i-1}(x)=\frac{1}{\left(r_{i}-1\right)!\prod_{\begin{subarray}{c}j=1\\ j\neq i\end{subarray}}^{n}\left(x_{i}-x_{j}\right)r_{j}}\cdot\frac{l(x)}{x-x_{i}},i=1,2,\ldots,n$
where
(7)

l(x)=\left(x-x_{1}\right)^{r_{1}}\left(x-x_{2}\right)^{r_{2}}\ldots\left(x-x_{nl}\right)^{r_{n}}.

In the considered case, formula (3) becomes
(8)

A[f]\approx\sum_{i=1}^{n}\sum_{j=0}^{r_{i}r^{-1}}c_{ij}f^{(j)}\left(x_{i}\right)

where

c_{i,j}=\mathbf{A}\left[l_{i},j\right],j=0,1,\ldots,r_{i}-1,i=1,2,\ldots,n

4.

The rest of formula (8) enjoys the important property that it cancels out for any polynomial of degree $p,p$ being given by formula (5).

In general if the linear functional $\mathbb{A}[f]$ is zero for any polynomial of degree $m^{2}$ ), but is nonzero for at least one polynomial of degree $m+1$ , it is said that it has the degree of accuracy $m$ This definition naturally extends to the cases $m=-1$ and $m=+\infty$ More precisely, the degree of accuracy is an integer $\geq-1$ or the wrong number $+\infty$ attached to the functionality $\mathrm{A}[f]$ and perfectly characterized by prosperity:
$1^{0}.m=-1$ , if $A[1]\neq 0$ ,
$2^{0}.\mathrm{A}[1]=\mathrm{A}[x]=\cdots=\mathrm{A}\left[x^{m}\right]=0,\mathrm{~A}\left[x^{n+1}\right]\neq 0$ , if $\mathrm{A}[1]=0$ and at least one of the numbers $\mathrm{A}\left[x^{i}\right],i=1,2,\ldots$ is different from zero,
$3^{0}.m=+\infty$ , if $\mathrm{A}\left[x^{i}\right]=0,i=0,1,\ldots$
Finally, we will say that the functional $\mathbb{A}[f]$ has the degree of accuracy at least $m$ , if it has the degree of accuracy $\geq m$ , so if it cancels out for any polynomial of degree $m$ .

For simplicity, we will say that an approximation formula (3) has the degree of accuracy $m$ respectively has the degree of accuracy at least $m$ , if the rest of this formula has the degree of accuracy $m$ respectively has the degree of accuracy at least $m$ .

With this convention we can say that formula (8) has the degree of accuracy at least $p$ .

We recall the following:
Theorem 1. If formula (3) has the degree of accuracy at least p, this formula necessarily coincides with (8).

I gave the proof of this property in another paper [9]. The proof there referred to a linear functional $A[f]$ particular, but it does not depend on the form of this functional.

Theorem 1 tells us that formula (8) plays a special role among the formulas (3) that are obtained by varying the coefficients of this formula. Formula (8) is, among all the formulas (3) corresponding to a functional $\mathrm{A}[f]$ and some nodes (2), given together with their respective multiplicity orders, the (unique) one that has the maximum degree of accuracy.
5. In general, the degree of accuracy of formula (8) is $p$ In particular cases, however, this degree of accuracy may be even greater than $p$ .

Definition. We will say that formula (8) is of Gaussian type if its degree of accuracy is at least $p+n$ .

⁰⁰footnotetext: 2. Through polynomial of degree

m

, we understand a polynomial of effective degree

\leqq m

A polynomial of degree 0 is a constant, and a polynomial of degree -1, identical to the zero polynomial,

For formula (8) to be Gaussian it is necessary and sufficient that its remainder $\mathrm{R}[f]$ to cancel out for any polynomial of degree $p+n$ .

A polynomial $P(x)$ of the degree $p+n$ is always of the form

P(x)=l(x)Q(x)+Q_{1}(x),

where $l(x)$ is the polynomial (7), $Q(x)$ a polynomial of degree $n-1$ and $Q_{1}(x)$ a polynomial of degree $p$ . Conversely, any polynomial of this form is of degree $p+n$ . Then $R[P]=R[lQ]$ and from (8) it immediately follows $R(lQ)=A[lQ]$ It follows therefore that the necessary and sufficient condition for formula (4) to be Gaussian is that we have

(10)

\mathrm{A}[l\mathrm{Q}]=0

whatever the polynomial is $\mathrm{Q}(x)$ of the degree $n-1$ .
Two functions $f,g$ for which we have $\AA [fg]=0$ can be called orthogonal to the functional $\mathrm{A}[f]$ We can therefore state:

Theorem 2. The necessary and sufficient condition for formula (8) to be of Gaussian type is that polynomial (7) is orthogonal to any polynomial of degree $n-1$ :

Orthogonality of the polynomial $l(x)$ with any polynomial of degree $n-1$ is equivalent to its orthogonality with $n$ polynomials of degree $n-1$ linearly independent. Such a system of $n$ polynomials is formed from the first $n$ power $1,x,x^{2},\ldots,x^{n-1}$ his/hers $x$ Another system of this kind is formed by the polynomials

\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{i-1}\right)\left(x-x_{i+1}\right)\ldots\left(x-x_{n}\right),i=1,2,\ldots,n,

(11)

because we assume that the nodes are distinct.
This last example shows us that a Gaussian formula (8) is nothing more than a formula of the form (8)

\mathrm{A}[f]\approx\sum_{i=1}^{n}\sum_{j=0}^{r_{i}}c_{i,j}^{\prime}f^{(j)}\left(x_{i}\right),

(12)

corresponding to nodes (2), but of multiplicity order $r_{i}+1$ (instead of $\left.r_{i}\right),i=1,2,\ldots,n$ respectively $\mathdollar 1$ in which we have $c^{\prime}{}_{i},r_{i}=0,i=1,2,\ldots.n$ . This observation is due in principle to AA Markov [4]. The property results from formulas (6); (9) corresponding to formula (12).
6. In the following we will assume that the functional $\mathrm{A}[f]$ , natural number $n$ and the orders of multiplicity $r_{1},r_{2},\ldots,r_{n}$ of the nodes are given.

The orthogonality condition, based on the observation above, can be interpreted in another way. We can look at

\phi=\phi\left(x_{1},x_{2},\ldots,x_{n}\right)=\mathrm{A}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{\prime}{}_{i}+1\right]

(13)

as a (polynomial) function of $x_{1},x_{2},\ldots,x_{n}$ . Then

\begin{gathered}-\frac{1}{r_{i}+1}\cdot\frac{\partial\phi}{\partial x_{i}}=\mathrm{A}\left[l(x)\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{i-1}\right)\left(x-x_{i+1}\right)\ldots\left(x-x_{n}\right)\right]\\ i=1,2,\ldots,n\end{gathered}

It follows that the nodes of a Gaussian formula (8) always form a solution of the algebraic system
(14)

\frac{\partial\phi}{\partial x_{i}}=0,\quad i=1,2,\ldots,n.

Any solution to this system, in which the values of the variables $x_{1},x_{2},\ldots,x_{n}$ are distinct, it gives us a Gaussian formula. However, system (14) does not always have such a (real) solution.

If multiple numbers $r_{i}$ are equal, we do not consider as distinct two Gaussian formulas that differ only by a permutation of the nodes having this multiplicity order. In other words, Gaussian formulas depend only on the distinct values of the multiplicity orders.

Based on this observation, one can easily transform system (14) into the equivalent jump from the point of view of searching for Gaussian-type formulas. Thus if $r_{1}^{\prime},r_{2}^{\prime},\ldots,r_{t}^{\prime}(1\leq t\leq n)$ are the distinct values of the multiplicity orders $r_{1},r_{2},\ldots,r_{n}$ , either $n_{i}$ number of nodes of multiplicity order $r_{i}^{\prime}$ $ $1\sigma_{1}^{(i)},\sigma_{2}^{(i)},\ldots,\sigma_{n_{i}}^{(i)}$ fundamental symmetric functifiles of these knots, $i=1,2,\ldots,t$ Then, from the point of view of searching for Gaussian formulas, system (14) is equivalent to

\frac{\partial\phi}{\partial\sigma_{j}^{(i)}}=0,j=1,2,\ldots,n_{i},i=1,2,\ldots,t

(15)

This system is as simple as (14) in the sense that the function $\phi$ is a polynomial with respect to $\sigma_{j}^{(i)},j=1,2,\ldots,n_{i},i=1,2,\ldots,t$ The equivalence of the systems (14', (15), in the specified sense, results from the fact that the functional determinant of the fundamental symmetric functions $\Sigma z_{1}z_{2}\ldots..z_{i},i=1,2,\ldots,k$ of variables $z_{1},z_{2},\ldots,z_{k}$ with respect to these variables is different from zero, for any system of different values of these variables (see e.g. [6]).

The same can be said about the system deduced from (14), replacing only in part the nodes corresponding to equal multiplicity orders by their fundamental symmetric functionals. The previous method can also be combined with some linear transformations of some or all of the variables. $x_{i}$ ,
7. We must note that for a given linear functional $\AA [f]$ and for a given system of multiplicity orders, there are not always Gaussian-type formulas.

We will say that the functional $\mathrm{A}[f]$ is of the order of positivity $k$ if $A\left[Q^{2}\right]>0$ , for any polynomial $Q(x)$ of the degree $k-1$ and non-identical null.

We can then observe that for a functional of order positivity $k\geq\frac{1}{2}(p+n+2)$ and if at least one of the multiplicity orders $r_{1},r_{2},\ldots,r_{n}$ is even, there is no Gaussian formula. Indeed, let us assume, for the sake of clarity, that $r_{i},r_{i+1},\ldots,r_{n}$ are even numbers, $0<i\leqq n$ , the others (if $1<i$ ) being odd. Then $l(x)\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{i-1}\right)$ is the square of a polynomial of degree $\frac{1}{2}(p+i)\leq k-1$ So we have $\mathbb{A}\left[l(x)\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{i-1}\right)\right]>0,(\mathbb{A}[l]>0$ , if $i=1)$ , which, based on Theorem 2 , proves the property.

On the contrary, we will see that if all the numbers $r_{1},r_{2},\ldots,r_{n}$ are odd, there is at least one Gaussian formula.

Formulas (14) immediately suggest us to examine the relative extrema of the function (13). A relative extremum reached by a system of different values of the variables $x_{1},x_{2},\ldots,x_{n}$ , we demonstrate the existence of at least one Gauss-type formula. We will see that based on this observation, this existence property holds in particular if $\mathbb{A}[f]$ is of the order of positivity $\geq\frac{1}{2}(p+n+1)$ and if all multiplicity orders are odd.

In the following we will study an extremum problem which, in particular, will give us the solution to the above problem. The properties obtained in this section should be considered as a generalization of the well-known extremal properties of orthogonal polynomials and their generalizations in the sense of G. Polya [5] and D. Jackson [2, 3].

§ 2. On a minimum problem.

8.

Let us consider, in particular, a linear functional of the form

\mathrm{A}[f]=\sum_{i=1}^{k}\lambda_{i}f\left(y_{i}\right),

(16)

where $k$ is a natural number, $y_{1},y_{2},,,,y_{k},k$ distinct points of the real axis and $\lambda_{1},\lambda_{2},\ldots,\lambda_{k}$ SYNTHESIS $k$ given positive numbers.

We will note with $P_{n}$ the set of (real) polynomials of degree $n$ formal $x^{n}+\cdots$ , so with its coefficient $x^{n}$ equal to 1.

They are given $t$ positive numbers $s_{1},s_{2},\ldots,s_{t}$ and $t$ natural numbers $n_{1},n_{2},\ldots,n_{t}$ so that each number $s_{i}$ What does a number correspond to? $n_{i}$ For the sake of brevity, we will call the numbers $s_{i}$ powers and numbers $n_{i}$ the respective degrees corresponding to these powers. These names are justified by the following.

Whether

\mu=\mu_{n_{1},n_{2},\ldots,n_{t}}^{\left(s_{1},s_{2},\ldots,s_{t}\right)}=\inf\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|s_{i}\right]

(17)

where the lower bound is relative to all polynomials $\pi_{i}\in P_{n_{i}}$ , $i=1,2,\ldots,t$ .

Any particular system of polynomials $\pi_{i}\in P_{n_{i}},i=1,2,\ldots,t$ for which $A\left[\prod_{i=1}^{t}\left|\pi_{i}\right|s_{i}\right]=\mu$ it will be called a system of minimizing polynomials or, for short, a minimizing system.

case $t=1$ is well known and has been examined in particular by D. Jackson [3].
9. In the case of $n=n_{1}+n_{2}+\cdots+n_{t}\geq k$ HAVE $\mu=0$ and because the system $\pi_{i}\in P_{n_{i}},i=1,2,\ldots,t$ to be minimizing it is necessary and sufficient that each point $y_{j},j=1,2,\ldots,k$ to be a root of at least one polynomial $\pi_{i}$ In the case of $n<k$ the results are less trivial and are given by the following:

Theorem 3. $\mathrm{A}[f]$ being a linear functional of the form (16) with the points $y_{i}$ distinct and with numbers $\lambda_{i}$ all positive and $s_{1},s_{2},\ldots,s_{t}$ a system of (positive) powers and $n_{1},n_{2},\ldots,n_{t}$ a system of degrees corresponding to given values, with their sum $n=n_{1}+n_{2}+\cdots+n_{t}<k$ :
$1^{0}$ There is at least one minimizing system.
$2^{0}$ If $\pi_{i}\in P_{ni},i=1,2,\ldots,t$ is a minimizing system, the polynomials $\pi_{i}$ they all have real roots.

If the powers $s_{1},s_{2},\ldots,s_{t}$ they are all $>1$ , then any minimizing system $\pi_{i}\in P_{n_{i}},i=1,2,\ldots,t$ also check the following properties:
$3^{0}$ . All roots of the polynomial
(18)

\pi=\pi(x)=\prod_{i=1}^{t}\pi_{i}

are distinct.
$4^{0}$ . We have ³ )
(19)

\left.\left.\mathrm{A}\left|\prod_{i=1}^{t}\right|\pi_{i}\right|^{s_{i}-1}(\mathrm{sg}\pi)\mathrm{Q}\right]=0,

whatever the polynomial is $Q(x)$ of the degree $n-1$ .

$5^{\circ}$ The roots of the polynomial (18) are separated by the points $y_{1},y_{2},\ldots,y_{k}$ .

We note that in the case of Theorem 3, $\mu>0$ .

10. For $1^{\circ}$ of Theorem 3 is proved by first showing that there exists a positive number $a$ so that if at least one of the coefficients of the polynomials $\pi_{i},i=1,2,\ldots,t$ is $\geq a$ in absolute value, we have

(20)

\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\right]>\mu.

This property results from the following three lemmas:
Lemma 1. If $m<k$ and $\pi\in P_{m}$ , there is a positive number $b_{m}$ so that
(21)

|\pi|\geq b_{m}

at least $k-m$ from the points $y_{i},i=1,2,\ldots,k$ .
It is for this
(22)

\mathrm{E}\left(z_{1},z_{2},\ldots,z_{m+1}\right)

the best approximation, in Chebisev's sense, of $x^{n}$ in the polis, we are called by the degree $m-1$ on distinct points $z_{1},z_{2},\ldots,z_{m+1}$ . The value of (22) is well known [13] but there is no need to reproduce it here. We only note that this number is positive ⁴ ).

Let's take $b_{m}=\min\mathrm{E}\left(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m+1}}\right)$ , where the minimum refers to all combinations $i_{1},i_{2},\ldots,i_{m+1}$ Cite $m+1$ of the indices $1,2,\ldots,k$ . From the properties of best approximation polynomials and the definition of the number $b_{m}$ it turns out that among the first $m+1$ puncture $y_{i}$ there is one, either $y_{j}$ which $|\pi|\geq b_{m}$ Leaving aside the point $y_{j1}$ , among the first $m+1$ puncture $y_{i}$ left, there is one, either $y_{j_{2}}$ , which $|\pi|\geqq b_{m}$ . We leave aside the point $y_{j2}$ and we continue the process. In this way we determine the points $y_{j1}$ , $y_{j_{2},\ldots,}y_{j_{m+1}}$ on which inequality (21) is verified.

⁰⁰footnotetext: 3. We have

sgz=-1,0

resp, 1 as

z<

=

respectively.

>0

s

is a positive and even integer, we have

|z|s-1\mathrm{sg}z=z^{s-1},|z|s-1=z^{s-1}

z

, whatever it is

z

. 4. If

\epsilon=\min\left|z_{i}-z_{j}\right|

, we have

\mathrm{E}\left(z_{1},z_{2},\ldots,z_{m+1}\right)>\frac{\epsilon^{m}}{m+1}

We notice that the number $b_{m}$ does not depend on the polynomial $\pi$ .
Lemma 2. If $\pi\in P_{m}$ and if we have $|\pi|\leqq\mathrm{M}$ , on $m$ distinct points $z_{1},z_{2},\ldots,z_{m}$ , then there is a positive number $\mathrm{F}\left(z_{1},z_{2},\ldots,z_{m}\right)$ so that all the coefficients of the polynomial $\pi$ to be $<\mathrm{F}\left(z_{1},z_{2},\ldots,z_{m}\right)\mathrm{M}$ , in absolute value $\check{a}$

The property is well known and we can dispense with giving its proof here ⁵ ).

Lemma 3. If $\pi_{i}\in P_{ni},i=1,2,\ldots,t,n=n_{1}+n_{2}\ldots+n<k$ and if

\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\leqq\mathrm{~N},

(23)

on the points $y_{1},y_{2},\ldots,y_{k}$ , there is a positive number $c$ , independent of the polynomials $\pi_{i},i=1,2,\ldots,t$ , so that all the coefficients of these polynomials are $<c\mathrm{~N}$ in absolute value.

The proof can be done by complete induction on $t$ For $t=1$ the property is true because then (23) becomes $\left|\tau_{1}\right|^{\mid s_{1}}\leq N$
and, based on Lemma 2, the coefficient of the polynomial $\pi_{1}$ ARE $<\mathrm{F}\left(y_{1},y_{2},\ldots,y_{n_{1}}\right)\mathrm{N}^{\overline{s_{1}}}$ in absolute value.

Suppose the property is true for $t-1(t>1)$ powers and demonstrate it for $t$ powers. Let us therefore assume that we have (23). Based on Lemma 1, let $y_{j_{1}},y_{j_{2}},\ldots,y_{j_{k-n_{1}}},k-n_{1}$ between the points $y_{i}$ which we have $\left|\pi_{1}\right|\geqq b_{n_{1}}$ . Then $k-n_{1}>n_{2}+n_{3}+\cdots+n_{t}$ and on these points we have $\underset{i=2}{\stackrel{{\scriptstyle t}}}\left|\pi_{i}\right|{}^{s}i\leqq Nb_{n_{1}}^{-s_{1}}$ It follows that there was a number $c^{\prime}$ so that all the coefficients of the polynomials $\pi_{2},\pi_{3},\ldots,\pi_{t}$ to be $<c^{\prime}b_{n_{1}}^{-s_{1}}\mathrm{~N}$ , in absolute value. Similarly, we prove that there is a number $c^{\prime\prime}$ so that all the coefficients of the polynomials $\pi_{1},\pi_{3},\pi_{4},\ldots,\pi_{t}$ to be $<c^{\prime\prime}b_{n_{2}}^{-s_{2}}\mathrm{~N}$ , in absolute value. If $c=\max\left(c^{\prime}b_{n_{1}}^{-s_{1}},c^{\prime\prime}b_{n_{2}}^{-s_{2}}\right)$ , we see that the property is true for $t$ powers. With this, Lemma 3 is proven.
11. Let us return to pt. $1^{0}$ of theorem 3.

We have obviously

\mu\leqq\mathrm{A}\left[|x|^{\left.\mid n_{1}s_{1}+n_{2}s_{2}+\cdots+n_{t}s_{t}\right]=\mathrm{T}.}\right.

Let us now assume that

\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\leqq\mathrm{~N}_{1}=\frac{\mathrm{T}}{\min\left(\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\right)}

and either $c_{1}$ , the number $c$ corresponding to $\mathrm{N}=\mathrm{N}_{1}$ from Lemma 3.
If then $a=c_{1}N_{1}$ and if at least one of the coefficients of the polynomials $\pi_{1},\pi_{2},\ldots,\pi_{t}$ is $\geqq a$ in absolute value, we have $\prod_{i=1}\left|\pi_{i}\right|s_{i}>\mathrm{N}_{1}$ on
5) If $\delta=\max_{i}\left|z_{i}\right|$ , we have

\mathrm{F}\left(z_{1},z_{2},\ldots,z_{m}\right)<\frac{\mathrm{M}}{\mathrm{E}\left(z_{1},z_{2},\ldots,z_{u}u\right)}+(\delta+1)^{m}.

at least one of the points $y_{i}$ We then have $\left.\left.A\left|\prod_{i=1}^{t}\right|\pi_{i}\right|^{s_{i}}\right]>T\geq\mu$ and ines
the equality (20) is verified. equality (20) is verified.

By well-known reasoning we can now demonstrate the existence of at least one minimizing system.

From his definition $\mu$ and from the previous properties, it follows that we can find an infinite series of polynomial systems $\pi_{i}{}^{(m)}\in P_{n_{i}},i=1,2,\ldots,t,m=1,2,\ldots$ so that, on the one hand, all the coefficients of these polynomials are $<a$ in absolute value sí, on the other hand, if for abbreviation we put $\pi^{(m)}=\prod_{i=1}^{t}\left|\pi_{i}^{(m)}\right|s^{s}$ to have

\lim_{m\rightarrow\infty}\mathbb{A}\left[\Pi^{(m)}\right]=\mu.

We can then extract from the string $\left(\Pi^{(m)}\right)_{m=1}^{\infty}$ a partial string $\left(\Pi^{(vm)}\right)_{m=1}^{\infty}$ so that

\lim_{n\rightarrow\infty}\pi_{i}^{(\gamma m)}=\pi_{i}^{*}\in P_{n_{i}},\quad i=1,2,\ldots,t

uniformly in any finite interval.
It then follows that the polynomials $\pi_{i}^{*},i=1,2,\ldots,t$ form a minimizing system.

With this for $1^{0}$ of Theorem 3 is proved.
12. Let $\pi_{i},i=1,2,\ldots,t$ a system of minimizing polynomials. Let us also assume that $1\leq u<t$ , where $u$ is a natural number and is a polynomial $\pi_{1},\pi_{2},\ldots,\pi_{u}$ decomposes into the product of two real factors $\pi_{i}=\pi_{i}^{\prime}\pi_{i}^{\prime\prime}$ , $i=1,2,\ldots,u$ so that $\pi_{i}^{\prime}\in P_{n^{\prime}i}\pi_{i}^{\prime\prime}\in P_{n^{\prime\prime}i}$ so $\mathdollar\stackreln_{i}^{\prime}+n_{i}^{\prime\prime}=n_{i}$ We can assume $n_{i}^{\prime}>0,i=1,2,\ldots,u$ and $\mathrm{c}^{*}$ , in particular, the factor $\pi_{i}^{\prime\prime}$ can also be reduced to 1 (then $n_{i}^{\prime\prime}=0,n_{i}^{\prime}=n_{i}$ ).

Let us consider the linear functional

\mathrm{A}_{1}[f]=\mathrm{A}\left[\left(\prod_{i=1}^{n}\left|\pi_{i}^{\prime\prime}\right|s_{i},\prod_{i=u+1}^{n}\left|\pi_{i}\right|s_{i}\right)f\right]

(24)

This functional is of the form $A_{1}[f]=\sum_{j=1}^{k}\lambda_{j}^{\prime}f\left(y_{j}\right)$ , where

\lambda_{j}^{\prime}=\lambda_{j}\prod_{i=1}^{u}\left|\pi_{i}^{\prime\prime}\left(y_{j}\right)\right|^{s_{i}}\cdot\prod_{i=u+1}^{n}\left|\pi_{i}\left(y_{j}\right)\right|^{s_{i}},j=1,2,\ldots,k

It is seen that at most $n_{1}^{\prime\prime}+n_{2}^{\prime\prime}+\cdots+n_{u}^{\prime\prime}+n_{u+1}n_{u+2}+\cdots+n_{t}$ coefficients $\lambda_{j}^{\prime}$ is canceled and at least $k^{\prime}=k-\left(n+n_{2}+\cdots+n_{t}^{\prime}+n_{1}^{\prime}+n_{2}^{\prime}+\cdots+\right.+n_{u}^{\prime}>n_{1}^{\prime}+n_{2}^{\prime}+\cdots+n_{u}^{\prime}$ are positive. For $1^{0}$ of Theorem 3 applies to the functional (24) and the system of polynomials $\pi_{1}^{\prime},\pi_{2}^{\prime},\ldots,\pi_{u}^{\prime}$ necessarily coincides ^with a minimizing system for the functional (24), for the powers $s_{1},s_{2},\ldots,s_{u}$ to which the degrees correspond respectively $n_{1}^{\prime},n_{2}^{\prime},\ldots,n_{u}^{\prime}$ To show this, let us assume the opposite and then let $\pi_{1}^{*},\pi_{c}^{*},\ldots,\pi_{u}^{*}$ a minimizing system corresponding to the functionality $\mathrm{A}_{1}[f]$ We have
$\mathrm{A}\left[\prod_{i=1}^{u}\left|\pi_{i}^{*}\pi_{i}^{\prime\prime}\right|^{s_{i}}\prod_{i=u+1}^{t}\left|\pi_{i}\right|^{s_{i}}\right]=\mathrm{A}_{1}\left[\prod_{i=1}^{u}\left|\pi_{i}^{*}\right|^{s_{i}}\right]<\mathrm{A}_{1}\left[\prod_{i=1}^{u}\left|\pi_{i}^{\prime}\right|^{s_{i}}\right]=\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\right]$
so

\mathrm{A}\left[\prod_{i=1}^{u}\left|\pi_{i}^{\star}\pi_{i}^{\prime\prime}\right|s_{i}\prod_{i=u+1}^{t}\left|\pi_{i}\right|^{s}i\right]<\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|s_{i}\right]

which contradicts the hypothesis that $\pi_{i},i=1,2,\ldots,t$ it is a minimizing system.

13. Based on the above observations, to demonstrate for $2^{\circ}$ of Theorem 3 it is enough to assume $t=1$ , which simplifies the reasoning.

Be it then $\pi\in P_{n}$ a minimizing polynomial. Suppose that $\pi$ would not have all real roots. Then this polynomial has a real factor of the form $(x-a)^{2}+b^{2}$ , where $b\neq 0$ .

Let's put

\pi(x)=\bigl[(x-a)^{2}+b^{2}\bigr]Q(x),\quad\pi_{1}(x)=(x-a)^{2}Q(x).

Then $\pi_{1}\in P_{n}$ and $|\pi_{1}|\leq|\pi|$ for anything $x$ But the strict inequality $|\pi_{1}|<|\pi|$ is checked on at least one of the points $y_{i}$ .

Therefore,

A\bigl[|\pi_{1}|_{1}\bigr]<A\bigl[|\pi|_{1}\bigr],

which contradicts the hypothesis that $\pi$ is a minimizing polynomial.

With this, for $2^{0}$ of Theorem 3 is proven.
14. To prove pt. $3^{0}$ of theorem 3, based on what was established in no. 12, 13, it is enough to consider only the case $t=2,n_{1}=n_{2}=1$ . If then $\pi_{1},\pi_{2}$ is a minimizing system, we must show that $\pi_{1}\neq\pi_{2}$ . Let's assume the opposite, so that we would have $\pi_{1}=\pi_{2}=\pi$ and be $\psi(\epsilon)=\mathrm{A}\left[\left|\pi+s_{2}\epsilon\right|{}^{s_{1}}\left|\pi-s_{1}\epsilon\right|s_{2}\right]$ . Then $\psi(\epsilon)$ is a continuous function and has a continuous derivative in $\epsilon\left(s_{1},s_{2}>1\right)$
We have
(25 ) $\frac{d\psi}{d\epsilon}=-s_{1}s_{2}\left(s_{1}+s_{2}\right)\epsilon\mathrm{A}\left[\left|\pi+s_{2}\epsilon\right|^{s_{1}-1}\left|\pi-s_{1}\epsilon\right|^{s_{2}-1}sg\left(\pi+s_{2}\epsilon\right)\left(\pi-s_{1}\epsilon\right)\right]$

But

\mathrm{A}\|\pi+s_{2}\epsilon\left|s_{1}-1\right|\pi-\left.s_{1}\epsilon\right|^{s_{2}-1}\operatorname{sg}\left(\pi+s_{2}\epsilon\right)\left(\pi-s_{1}\epsilon\right)\mid>0,

for $|\epsilon|$ quite small, since the first term is a continuous function of $\epsilon$ which for $\epsilon=0$ it reduces to $A\left[|\pi|s_{1}+s_{2}-2\right]>0$ From (25) it follows that sg $\frac{d\psi}{d\epsilon}=-$ sg $\epsilon$ , for $|\epsilon|$ quite small, which shows us that $\Psi(\epsilon)$ has a relatively strict maximum for $\epsilon=0$ For $|\epsilon|$ quite small but $\neq 0$ we have therefore

\mathbb{A}\left[\left|\pi+s_{2}\in\right|^{s_{1}}\left|\pi-s_{1}\in\right|^{s_{2}}\right]<\mathbb{A}\left[|\pi|{}^{s_{1}}+s_{2}\right]

which contradicts the hypothesis that $\pi_{1}=\pi_{2}=\pi$ is a minimizing system. With this we have demonstrated that $\pi_{1}\neq\pi_{2}$ and therefore for $3^{0}$ of Theorem 3.
15. Let $x_{1},x_{2},\ldots,x_{n}$ roots of the polynomial $\pi_{1}\pi_{2}\ldots\pi_{t}$ , then
(26)

\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s}i\right]=\mathrm{A}\left[\prod_{i=1}^{n}\left|x-x_{i}\right|^{s^{\prime}i}\right]

is a continuous function of $x_{1},x_{2},\ldots,x_{n}$ . Based on the previous results, at any point where the lower bound (17) is reached, we have a relative minimum of the function. If the powers $s_{1},s_{2},\ldots,s_{t}$ they are all $>1$ , these minima are reached only for different values of the variables $x_{i}$ . In this case,
however, the function (26) is differentiable and therefore, at these points, the first-order partial derivatives of the function (26) are zero. We have ⁶ )

		$\displaystyle\frac{\partial}{\partial x_{i}}\mathrm{~A}\left[\prod_{i=1}^{t}\left\|\pi_{i}\right\|^{si}\right]=s_{i}^{\prime}\mathrm{A}\left[\prod_{\begin{subarray}{c}j=1\\ j\neq i\end{subarray}}^{n}\left\|x-x_{j}\right\|^{s^{\prime}j}\cdot\left\|x-x_{i}\right\|^{s^{\prime}i}-1\operatorname{sg}\left(x-x_{i}\right)\right]=$
	$\displaystyle=$	$\displaystyle s_{i}^{\prime}\mathrm{A}\left[\prod_{i=1}^{t}\left\|\pi_{i}\right\|^{s}i^{-1}\cdot\operatorname{sg}\left(\prod_{i=1}^{t}\pi_{i}\right)\cdot\left(x-x_{1}\right)\ldots\left(x-x_{i-1}\right)\left(x-x_{i+1}\right)\ldots\left(x-x_{n}\right)\right],$

where does it result from? $4^{0}$ of theorem 3, noting that under the conditions here the polynomials (i1) are linearly independent.
16. The statement for $5^{0}$ of theorem 3 constitutes, in a more complete form, a reciprocal of the previous properties in the sense that for $3^{0}$ results from $4^{0}$ . In fact, the property is somewhat more general and can be stated as follows:

Theorem 4. $\mathrm{A}[f]$ being a linear functional of the form (16) with the points $y_{i}$ distinct and with numbers $\lambda_{i}$ all positive, and $s_{1},s_{2},\ldots,s_{i}$ a system of all powers $>1$ and $n_{1},n_{2},\ldots,n_{t}$ a system of corresponding degrees, with their sum $n<k$ , if the polynomials $\pi_{i}\in P_{n_{i}},i=1,2,\ldots,t$ verify equality (19) for any polynomial $\mathrm{Q}(x)$ of the degree $n-1$ , then all the roots of the polynomial (18) are real, distinct and separated from the points $y_{i}$ , $i=1,2,\ldots,k$ .

The separation property of the statement means that if we assume the points $y_{i}$ arranged in ascending order, so $y_{1}<y_{2}<\cdots<y_{k}$ , we have $\pi\left(y_{1}\right)\neq 0,\pi\left(y_{k}\right)\neq 0$ and the string

\pi\left(y_{1}\right),\pi\left(y_{2}\right)\ldots.,\pi\left(y_{k}\right)

(27)

presents (after removing any zero terms) exactly $n$ sign variations.

We first observe that the polynomial $\pi$ being of the degree $n$ , the string (27) represents at most $n$ sign variations. In addition, if $\pi\left(y_{1}\right)=0,\pi\left(y_{k}\right)\neq 0$ or if $\pi\left(y_{1}\right)\neq 0,\pi\left(y_{k}\right)=0$ at most $n-1$ , and if $\pi\left(y_{1}\right)=\pi\left(y_{k}\right)=0$ at most $n-2$ variations of sign. Finally, we observe that from $n<k$ it follows that at least one term of the sequence (27) is different from zero.

Let us now suppose that the string would only present $n^{\prime}<n$ sign variations and either

\pi\left(y_{j_{1}}\right),\pi\left(y_{j_{2}}\right),\ldots,\pi\left(y_{j_{n^{\prime}+1}}\right),j_{1}<j_{2}<\cdots<j_{n^{\prime}+1}

a substring of (27) that represents exactly $n^{\prime}$ sign variations. Let $i_{v}$ the smallest natural number such that $\pi\left(y_{jv}\right)\pi\left(y_{jv+iv}\right)<0$ . Thus $1\leq i_{v}\leq j_{v+1}-j_{v}$ , $v=1,2,\ldots,n^{\prime}$ Let's take the points. $\xi_{1},\xi_{2},\ldots,\xi_{n^{\prime}}$ so that $y_{j_{v}}<\xi_{v}<y_{jv+i_{v}}$ , $v=1,2,\ldots,n^{\prime}$ and consider the polynomial $\left.\mathrm{Q}(x)=\pi\left(y_{jn^{\prime}+1}\right)\left(x-\xi_{1}\right)x-\xi_{2}\right)\ldots\left(x-\xi_{n^{\prime}}\right)$ which is the degree $n^{\prime}\leq n-1$ and which does not cancel out at any of the points $y_{i}$ From the way the points were chosen $\xi_{v}$ , it follows that we have $si\pi\left(y_{j}\right)\mathrm{Q}\left(y_{t}\right)\geq 0,i=1.2,\ldots,k$ , and then we have

6.

we
$\frac{d|x|s}{dx}=s|x|s-1$ sgx for $s>1$ and $\frac{d^{2}|x|s}{dx^{2}}=s(s-1)|x|s-2$ for $s\geq 2$

\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}-1}(\mathrm{sg}\pi)\mathrm{Q}\right]=\mathrm{A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|s_{i}-1|\mathrm{Q}|\right]>0

in contradiction with equality (19).
With this, Theorem 4 is proven. It generalizes a property established long ago for $t=1,s_{1}=2$ [7].
17. Theorem 3 and the previous results show us that the minimum problem treated always returns to the particular case when the degrees corresponding to the powers are all equal to 1.

If for abbreviation we write $\mathrm{cu}\mu^{s_{1}},s_{2},\ldots,s_{t}$ the number (17) when all the degrees $n_{1},n_{2},\ldots,n_{t}$ are equal to 1, we have

\mu^{s_{1}},s_{2},\ldots,s_{t}<\mu^{s_{2}}+s_{2},s_{3},s_{4},\ldots,s_{t}\quad(t>3)

and in particular $\mu^{s_{1}},s_{2},\ldots,s_{t}<\mu^{s_{1}}+s_{2}+\cdots+s_{t}$
In particular cases $t=1$ and $s_{1}=s_{2}=\cdots=s_{t}$ are equivalent in the previous sense.

To specify the uniqueness of the minimizing system, we must say that two minimizing polynomial systems in which, for each group of equal powers, the product of the polynomials is not considered distinct are not considered. $\pi_{i}$ is the same.

It is known that the minimizing system is unique if the powers are equal and $>1$ [3].

If the powers $s_{i}$ are not all equal, uniqueness no longer occurs in general, as will result from the examples in $\mathdollar 4$ These examples also show us that the property expressed by pt. $4^{0}$ of Theorem 3 does not characterize minimizing systems, in other words, there are also polynomials (18) for which the orthogonality property $4^{0}$ of Theorem 3 is verified but which are not formed with a minimizing system.

If the powers $s_{1},s_{2},\ldots,s_{t}$ they are all $\geq 2$ , so and $s_{i}^{\prime}\geq 2,i=1,2,\ldots,n$ we can easily prove that any system of polynomials $\pi_{i},i=1,2,\ldots,t$ which verifies the property $4^{\circ}$ of Theorem 3 , so in particular the minimizing polynomials, correspond to strict relative minima of the function (26). Indeed, in this case, the function (26) also has second-order partial derivatives and we have

\begin{gathered}\frac{\partial^{2}}{\partial x_{i}^{2}}\mathrm{~A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\right]=s_{i}^{\prime}\left(s_{i}^{\prime}-1\right)\mathrm{A}\left[\left(\prod_{\begin{subarray}{c}j=1\\ j\neq i\end{subarray}}^{n}\left|x-x_{i}\right|^{s^{\prime}j}\right)\left|x-x_{i}\right|^{s^{\prime}i-2}\right]>0\\ i=1,2\ldots.,n\end{gathered}

(28)

\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\mathrm{~A}\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}}\right]=

$=s_{i}^{\prime}s_{j}^{\prime}\mathrm{A}\left[\left(\prod_{\begin{subarray}{c}v=1\\ v\neq i,v\neq j\end{subarray}}^{n}\left|x-x_{v}\right|^{s^{\prime}v}\right)\left|x-x_{i}\right|^{s^{\prime}{}_{i}-1}\left|x-x_{j}\right|^{s^{\prime}j^{-1}}\mathrm{sg}\left(x-x_{i}\right)\left(x-x_{j}\right)\right]=$
$=s_{i}^{\prime}s_{j}^{\prime}\mathrm{A}\mid\left(\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}-1}\right)\left(\mathrm{sg}\prod_{i=1}^{t}\pi_{i}\right)\left(x-x_{1}\right)\ldots\left(x-x_{i-1}\right)\left(x-x_{i+1}\right)\ldots\left(x-xj_{-1}\right)$ .
$\left.\cdot\left(x-x_{j+1}\right)\ldots\left(x-x_{n}\right)\right]\quad i=1,2,\ldots,j-1,j=2,3,\ldots,n$ .

Based on the orthogonality property (19) at the points considered, all derivatives (28) are zero and the stated property results. In the case when $n>k$ , it is obvious that, in the above sense, there are an infinity of minimizing systems.

If $n=k$ , uniqueness does not occur, in the above sense, unless the powers are equal.

It is noteworthy that for $4^{\circ}$ of Theorem 3 also holds in the case $n\geq k$ , even with its reciprocal, in the sense that, if (19) holds for any polynomial $Q(x)$ of the degree $n-1$ , the system $\pi_{i},i=1,2,\ldots,t$ it is minimizing.

Indeed, let us assume that the system would not be minimizing, so that the polynomial (18) would not vanish at all points $y_{i}$ Either, for fixing ideas, $\pi(y_{k})\neq 0$ We then have…

\begin{gathered}A\left[\prod_{i=1}^{t}\left|\pi_{i}\right|^{s_{i}-1}\cdot(\operatorname{sg}\pi)\left(x-y_{1}\right)\left(x-y_{2}\right)\ldots\left(x-y_{k-1}\right)\right]=\\ =\lambda_{k}\prod_{i=1}^{t}\left|\pi_{i}\left(y_{k}\right)\right|^{s_{i}-1}\operatorname{sg}\left(\pi\left(y_{k}\right)\right)\cdot\left(y_{k}-y_{1}\right)\left(y_{k}-y_{2}\right)\ldots\left(y_{k}-y_{k-1}\right)\neq 0\end{gathered}

which contradicts equality (19).

§ 3. The existence of Gauss-type formulas.

18.

We first have the following:

Theorem 5. For any linear functional $\mathrm{A}[f]$ of the form (16), with the points $y_{i}$ distinct and with numbers $\lambda_{i}$ all positive, relative to any natural number $n<k$ and to any multiplicity order system $r_{1},r_{2},\ldots.r_{n}$ consisting of odd numbers, there is at least one Gaussian formula (8).

The hypothesis that the numbers $r_{1},r_{2},\ldots,r_{n}$ that they are all odd is essential, as can easily be seen from considerations analogous to those made in No. 7.

Theorem 5 follows from Theorem 3. To see this, it is enough to take $t=n$ , the powers $s_{1},s_{2},\ldots,s_{n}(\geq 2)$ respectively equal to $r_{1}+1,r_{2}+1,\ldots,r_{n}+1$ and the corresponding degrees all equal to 1. If (2) are the roots of the polynomial (18) corresponding to a minimizing system, we have

l(x)=\prod_{i=1}^{n}\left|x-x_{i}\right|^{s_{i}-1}\operatorname{sg}\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{n}\right)

and condition (19) reduces to the orthogonality of the polynomial $l(x)$ with any polynomial of degree $n-1$ .

In this way, each minimizing system corresponds to a Gaussian formula (8).
19. Let us return to a linear functional $\mathbb{A}[f]$ in order of positivity $k$ , as it was defined in no. 7. Obviously, if $\mathbb{A}[f]$ has the order of positivity $k$ , it also has the order of positivity $k^{\prime}$ for anything $k^{\prime}<k$ .

If we introduce the moments

\alpha_{i}=\mathrm{A}\left[x^{i}\right],i=0,1,\ldots

(29)

and Hankel determinants

A_{j}=\left\|\alpha_{\lambda+\mu}\right\|_{\lambda,\mu=0,1,\ldots,j},\quad j=0,1,\ldots

appropriate, the necessary and sufficient condition that the functional $A[f]$ to have the order of positivity $k$ it is so that we have…

A_{j}>0,j=0,1,\ldots,k-1,

or, which is equivalent, as the quadratic form $\sum_{i,j=0}^{k-1}\alpha_{i+j}\xi_{i}\xi_{j}$ to be definite and positive.

It is clear that in general the determination of Gaussian-type formulas depends only on the mutual ratios of the first $p+n+1$ MOMENTS $\alpha_{i}$ , $i=0,1,\ldots,p+n$ his/hers $\mathrm{A}[f]$ More precisely, in determining Gaussian formulas, one can abstract from a linear transformation of the variable $x$ and by a non-zero constant factor of the functional $\mathrm{A}[f]$ . Otherwise, this observation is generally valid for formulas of the form (3) which, through the indicated transforms ^∗ s, retain their form and degree of accuracy.

If $\mathrm{A}[f]$ is a linear functional of positive order $k$ , there is a polynomial $\rho_{k}\in P_{k}$ and only one that is orthogonal to any polynomial of degree $k-1$ This is the orthogonal polynomial of degree $k$ attached functionally $\mathrm{A}[f]^{7}$ ).

The polynomial $\rho_{k}$ has all real and distinct roots. Indeed, otherwise, this polynomial should have a divisor of the form $(x-a)^{2}+b^{2}$ ( $a,b$ real). If then $p_{k}=\left[(x-a)^{2}+b^{2}\right]\mathrm{Q}$ , Q is a polynomial of degree $k-2$ and we have $\AA \left[\rho_{k}\mathrm{Q}\right]=\mathrm{A}\left[((x-a)\mathrm{Q})^{2}\right]+b^{2}\mathrm{~A}\left[\mathrm{Q}^{2}\right]>0$ , which contradicts the orthogonality property.

It is clear that the polynomial $\rho_{k}$ depends only on the first $2k$ MOMENTS $\alpha_{i},i=0,1,\ldots,2k-1$ his/hers $\mathrm{A}[f]$ .

In particular, a functional of the form (16), in which $y_{i}$ are distinct and $\lambda_{i}$ positive, has the degree of positivity $k$ . In this case $\rho_{k}(x)=\left(x-y_{1}\right)\left(x-y_{2}\right)\ldots\left(x-y_{k}\right)$ .

But we also have a kind of reciprocal of this property in the following sense. If $y_{i},i=1,2,\ldots,k$ are the roots of the polynomial $\rho_{k}$ and the moments (29) verify the inequalities (30), there is a linear functional $\AA ^{(k)}[f]$ of the form (16), with all coefficients $\lambda_{i}$ positive and so that we have

\alpha_{i}=\mathrm{A}^{(t)}\left[x^{i}\right],i=0,1,\ldots,2k-1.

(31)

Indeed, any polynomial $\mathrm{Q}(x)$ of the degree $k$ is of the form

\left.\mathrm{Q}(x)=a\rho_{k}(x)+\sum_{i=1}^{k}\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\rho^{\prime}{}_{k}\left(y_{i}\right)}\mathrm{Q}y_{i}\right)

where the constant $a$ is equal to zero if and only if $\mathrm{Q}(x)$ is of the degree $k-1$ If $\mathrm{P}(x)$ is a polynomial of degree $2k-1$ , he is still $=$ one the product of two (real) polynomials $\mathrm{Q}(x),\mathrm{Q}_{1}(x)$ , first degree $k$ and the second in rank $k-1$ So we have

		$\displaystyle\mathrm{P}(x)=\mathrm{Q}(x)\mathrm{Q}_{1}(x)=\left[a\rho_{k}(x)+\sum_{i=1}^{k}\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\rho_{k}^{\prime}\left(y_{i}\right)}\mathrm{Q}\left(y_{i}\right)\right]$
		$\displaystyle{\left[\sum_{i=1}^{k}\frac{\rho_{k}(z)}{\left(x-y_{i}\right)\rho_{k}^{\prime}\left(y_{i}\right)}\mathrm{Q}_{1}\left(y_{i}\right)\right]=a\sum_{i=1}^{k}\rho_{k}(x)\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\rho_{k}^{\prime}\left(y_{i}\right)}\mathrm{Q}_{1}\left(y_{i}\right)+}$

⁰⁰footnotetext: 7. Existence and uniqueness of the polynomial

\rho_{k}

results only from

A_{k-1}\neq 0

A_{k-1}=0

such a polynomial either does not exist or it is not uniquely determined.

\begin{gathered}+\sum_{i=1}^{k}\left(\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\rho_{k}^{\prime}y_{i}}\right)^{2}\mathrm{P}\left(y_{i}\right)+\\ +\sum_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{k}\rho_{k}(x)\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\left(x-y_{j}\right)\rho_{k}^{\prime}\left(y_{i}\right)\rho_{k}^{\prime}\left(y_{j}\right)}\mathrm{Q}\left(y_{i}\right)\mathrm{Q}_{1}\left(y_{j}\right)\end{gathered}

If we take orthogonality into account, we deduce

\mathrm{A}[\mathrm{P}]=\mathrm{A}^{(k)}[\mathrm{P}]=\sum_{i=1}^{ib}\lambda_{i}\mathrm{P}\left(y_{i}\right)

where

\lambda_{i}=\mathrm{A}\left[\left(\frac{\rho_{k}(x)}{\left(x-y_{i}\right)\rho_{k}^{\prime}\left(y_{i}\right)}\right)^{2}\right]>0,i=1,2,\ldots,k

From this, in particular, the formulas (31) result.
From the previous analysis we recall:
Lemma 4. – If the linear functional $\mathrm{A}[f]$ has positivity ordinal $k$ , a linear functional can be found $\mathrm{A}^{(k)}[f]$ of the form (16) with all coefficients $\lambda_{i}$ positive and so that we have $\mathrm{A}[f]=\mathrm{A}^{(l)}[f]$ , for any polynomial of degree $2k-1$ .

It is easy to see that the linear functional $\AA ^{(k)}[f]$ is unique and is precisely the one determined above.
20. - We assume of course that if $\mathrm{A}[f]$ has the order of positís vita $k$ , intervalu1 I contains the roots of the orthogonal polynomial of degree $k$ attached to this functionality.

We can observe that if $a$ is a number such that $\mathrm{A}\left[(x-a)\mathrm{Q}^{2}\right]>0$ , for any polynomial Q of degree $k-1$ , the roots of the polynomial $\rho_{k}$ they are all $>a$ Indeed, if $\rho_{k}$ would have a root $x_{0}\leq a$ , then if we put $\rho_{k}(x)=\left(x-x_{0}\right)\mathrm{Q}(x)$ , we would have $\mathrm{A}\left[\rho_{k}\mathrm{Q}\right]=\mathrm{A}\left[(x-a)\mathrm{Q}^{2}\right]++\left(\alpha-x_{0}\right)\mathrm{A}\left[\mathrm{Q}^{2}\right]>0$ , which centers the orthogonality. It is also seen that if $\mathrm{A}\left[(x-b)\mathrm{Q}^{2}\right]<0$ , for any orthogonal Q , its nile $\rho_{k}$ they are all $<b$ .

Thus, for example, the classical functionals of Jacobí, Laguerre and Hermite

I^{(\alpha,\beta)}[f]=\int_{-1}^{+1}(1-x)^{\alpha}(1+x)^{\beta}f(x)dx\quad(\alpha,\beta>-1)

(32)

L^{(\alpha)}[f]=\int_{0}^{\infty}e^{-x}x^{\alpha}f(x)dx\quad(\alpha>-1)

(33)

H[f]=\int_{-\infty}^{+\infty}e^{-x^{2}}f(x)dx

(34)

have the order of positivity $k$ , for anything $k$ In the first case the roots of the orthogonal polynomials are in the interval ( $-1,1$ ), in the second in the interval $(0,\infty)$ and in the third case in the interval $(-\infty,+\infty)$ . In these cases it is
therefore sufficient to assume that I coincides with these intervals respectively.
In the case of positive linear functionals it follows in particular that the roots of orthogonal polynomials are inside the interval I .
21. - Returning to our problem, we can now prove

Theorem 6. - For any linear functional A [ $f$ ] which has the order of positivity $k$ , relative to any natural number and any multiplicity order system $r_{1},r_{2},\ldots,r_{n}$ consists only of odd numbers so that $k\geq\frac{1}{2}(p+n+1)$ , there is at least one Gaussian formula (8).

To prove this theorem it is enough to consider the functional $\AA ^{(k)}[f]$ determined by Lemma 4. Let (2) then be the nodes of a Gauss formula relative to the functional $\AA ^{(t)}[f]\quad\mathrm{O}$ such a formula exists because $k\geq n$ The polynomial $l(x)$ is orthogonal to any polynomial $\mathrm{Q}(x)$ of the degree $n-1$ functional side $\AA ^{(k)}[f]$ But the product $l(x)\mathrm{Q}(x)$ is of the degree $p+n\leq 2k-1$ , so $\mathrm{A}[l\mathrm{Q}]=\mathrm{A}^{(lk)}[l\mathrm{Q}]=0$ The polynomial $l(x)$ is therefore orthogonal to any polynomial of degree $n-1$ compared to the functional $A[f]$ , which, based on Theorem 2, proves the property.

It is also seen that all Gauss-type formulas related to the functional $A[f]$ are obtained in this way.

Equality $k=n$ it is only possible if $r_{1}=r_{2}=\cdots=r_{n}=1$ Then the Gaussian formula is unique and has as nodes precisely the roots of the polynomial $\rho_{k}$ orthogonally attached functionally $\mathrm{A}[f]$ Apart from this particular case, in the indicated hypotheses, the nodes of any Gaussian-type formula are separated by the roots of the orthogonal polynomial $\rho_{k}$ functional attachment $\mathrm{A}[f]$ .

We observe that, under the conditions of Theorem 6, we have
(35)

\begin{gathered}\mathrm{A}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}\right]=\\ =\mathrm{A}^{(k)}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}\right]+\mathrm{A}\left[x^{p+n+1}\right]-\mathrm{A}^{(k)}\left[x^{p+n+1}\right].\end{gathered}

It is seen that the minimum problem can be posed and solved on expression (35) from $\mathdollar 2$ , as in the case of functionals of the form (16). This is of course the minimum problem corresponding to the powers $r_{i}+1,i=1,2,\ldots,n$ and the respective degrees all equal to $1.{}^{8}$ ) The problem reduces, on the basis of equality (35), to a corresponding problem for a functional of the form (16). There are therefore, in particular, Gauss-type formulas corresponding to the minimizing system of this problem.

If $\dot{A}[f]$ has the order of positivity $k>\frac{1}{2}(p+n+1)$ , formula (35) can be replaced by

\mathrm{A}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}\right]=\mathrm{A}^{(k)}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}\right]

In particular, a positive lymph functional has a positivity order $k$ for anything $k$ and we therefore deduce

⁰⁰footnotetext: 8. The lower bound of this expression is no longer necessarily

\geq 0

Corollary 1. - For any positive linear functional, relative to any natural number n and to any multiplicity order system $r_{1},r_{2},\ldots,r_{n}$ consisting only of odd numbers, there is at least one Gauss-type formula.

In this case the nodes of a Gaussian formula are inside the interval I and are separated by the roots of any orthogonal polynomial of degree $k>\frac{1}{2}(p+n+1)$ attached to functionality.

In particular (32), (33), (34) are functionals of this kind.
The existence of Gaussian-type formulas for the case $r_{1}=r_{2}=\ldots=r_{n}=1\mathrm{~ms}$ par was proved by P. Turán [12].
22.- For the rest $\mathrm{R}[f]$ of the Gauss-type formulas ^{(8) , we have}

\mathrm{R}\left[x^{p+n+1}\right]=\mathrm{R}\left[l(x)\prod_{i=1}^{n}\left(x-x_{i}\right)\right]=\mathrm{A}\left[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}\right]

(36)

It follows that under the conditions of the theorem $6,\mathrm{R}\left[x^{p+n+1}\right]$ is the smallest, for and only for Gaussian formulas that come from minimizing systems.

If the functional $A[f]$ has the order of positivity $k>\frac{1}{2}(p+n+1)$ , all Gaussian formulas have the degree of accuracy equal to $p+n$ , because, in this case from (36) it follows that $\mathrm{R}\left[x^{p+n+1}\right]>0$ .

Based on an observation made in No. 5 and on the well-known expression of the remainder of the Lagrange-Hermite interpolation formula, the remainder $\mathrm{R}[f]$ of a Gaussian type formula (8) can be written
(37)

\mathrm{R}[f]=

$=\mathrm{A}[\prod_{i=1}^{n}\left(x-x_{i}\right)^{r_{i}+1}[\underbrace{x_{1},x_{1},\ldots,}_{r_{1}+1}x_{1},\underbrace{x_{2},x_{2},\ldots,x_{2}}_{r_{2}+1},\ldots,\underbrace{x_{n},x_{n},\ldots,x_{n}}_{r_{n}+1},x;f]]$
using a convenient notation of divided differences, which can be defined as follows:
Fie
(38)

\mathrm{V}\binom{f_{1},f_{2},\ldots,f_{m+1}}{z_{1},z_{2},\ldots,z_{m+1}}=\left\|f_{j}\left(z_{i}\right)\right\|_{i,j=1,2,\ldots,n+1}

determinant of function values $f_{i}=f_{i}(x),i=1,2,\ldots,m+1$ on the points $z_{1},z_{2},\ldots,z_{m+1}$ ( $i$ is the line index and $j$ of columns), provided that if a group of $v$ puncture $z_{i}$ are confused, those $v(>1)$ The corresponding lines contain the values of the functions and their primes. $v-1$ derivatives on this point. In particular

\mathrm{V}\left(z_{1},z_{2},\ldots,z_{m+}\right)=\mathrm{V}\binom{1,x,\ldots,x^{m}}{z_{1},z_{2},\ldots,z_{m+1}}

: is the Vandermonde determinant of the numbers $z_{1},z_{2},\ldots,z_{m+1}$ and

\left[z_{1},z_{2},\ldots z_{m+1};f\right]=\frac{\mathrm{V}\binom{1,x_{1},\ldots,x^{m-1},f}{z_{1},z_{2},\ldots,z_{m+1}}}{\mathrm{~V}\left(z_{1},z_{2},\ldots,z_{m+1}\right)}

is the divided difference (of the order $m$ ) of the function $f(x)$ on the nodes $z_{1},z_{2},\ldots$ , $z_{m+1}$ .

In the important case for applications, when $\mathrm{A}[f]$ is a positive linear functional, from (37) it follows that we have $\mathrm{R}[f]>0$ if $f(x)$ is a convex function of order $p+n$ . It is then known that we have [8],
(39)

\mathrm{R}[f]=\mathrm{R}\left[x^{p+n+1}\right]\mathrm{D}_{p+n+1}[f]

where, for brevity, we denote by $\mathrm{D}_{m}[f]$ a difference divided by order $m$ of the function $f(x)$ on $m+1$ convenient distinct nodes (depending on the function $f$ ) from within the interval I. These nodes can be chosen arbitrarily close to each other $[\mathbf{1,10}]$ .

If, in particular, the function $f(x)$ admits a derivative of the order $p+n+1$ , we have

\mathrm{R}[f]=\frac{\mathrm{R}\left[x^{p+n+1}\right]}{(p+n+1)!}f^{(p+n+1)}(\xi),

(40)

where $\xi$ is a convenient point inside the interval $I$ Also, if $f(x)$ has a derivative of order $p+n$ which verifies an ordinary Lipschitz condition with the constant $M$ , we have (41)

|\mathrm{R}[f]|\leqq\frac{\left|\mathrm{R}\left[x^{p+n+1}\right]\right|}{(p+n+1)!}\mathrm{M}

In formulas (39), (40), (41), the coefficient $\mathrm{R}\left[x^{p+n+1}\right]$ can be replaced with its value extracted from (36),

Finally, in this case we observe that in the sense of delimiting the remainder, the most precise Gauss-type formulas are those that correspond to minimizing systems.

In the particular case when $k=n$ and the functional is of the form (16), the Gaussian (unique) formula reduces to the trivial formula

\mathrm{A}[f]\approx\sum_{i=1}^{n}\lambda_{i}f\left(y_{i}\right)

with the remainder identically zero. The degree of accuracy of this formula is $+\infty$ .

§ 4. - Determination of some Gauss-type formulas

23.
- •
  
  To standardize notations, when it comes to a function: nalà lympha $\mathrm{F}[f]$ , we denote with the corresponding lowercase Greek letter, affected by the indices, the moments $\varphi_{i}=\mathrm{F}\left[x^{i}\right],i=0,1,\ldots,\mathrm{cu}$ corresponding uppercase Greek letter, affected by indices, Hankel's determinants $\phi_{i,j}=\left\|\varphi_{i+\lambda+\mu}\right\|\lambda,{}^{1}=0,1,\ldots,j,i,j=0,1,\ldots$ and, in particular, $\Phi_{j}=\Phi_{o,j}j=0,1,\ldots$

We also introduce the transformed moments $\varphi_{i,j}(\xi)=\mathrm{F}\left[(\xi-x)^{i}x^{j}\right]$ , $i,j=0,1,\ldots$ , where $\xi$ is an independent parameter of $x$ We then have

\varphi_{i,j}(\xi)=\sum_{v=0}^{i}(-1)^{v}\binom{i}{v}\xi^{i-v}\varphi_{v+j}

Hankel's determinant $\phi_{i,j}(\xi)=\left\|\varphi_{i+\lambda+\mu^{\mu},0}(\xi)\right\|_{\lambda,\mu_{a-0,1}\ldots,j}$ which is a polis nom in $\xi$ , can be brought, by elementary transformations of lines and columns, to other remarkable forms. Thus, we have ( $0\leq r\leq i$ ).
where we put

\delta_{i}^{v}(\xi)=\binom{i}{v}\xi^{i-v},v,i=0,1,\ldots,\left(\delta_{i}^{v}(\xi)=0,\text{ pentru }i<v\right)

In particular, for $r=i$ formula (42) becomes

\Phi_{i,j}(\xi)=\left|\begin{array}[]{lllll}\varphi_{0}&\varphi_{1}&\ldots&\ldots&\ldots\\ \varphi_{1}&\varphi_{2}&\ldots&\ldots&\varphi_{i+j}\\ \cdots&\cdots&\cdots&\varphi_{i+j+1}\\ \varphi_{j}&\varphi_{j+1}&&&\varphi_{i+2j}\\ \delta_{0}^{0}(\xi)&\delta_{1}^{0}(\xi)&\ldots&\ldots&\delta_{i+j}^{0}(\xi)\\ \delta_{0}^{1}(\xi)&\delta_{1}^{1}(\xi)&\ldots&\ldots&\delta_{i+j}^{1}(\xi)\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \delta_{0}^{t-1}(\xi)&\delta_{1}^{i-1}\left(\xi_{1}\right.&\ldots&\delta_{i+j}^{i-1}(\xi)\end{array}\right|

The formula is also valid for $r=0$ in the following form:

\Phi_{i,j}(\xi)=\left|\begin{array}[]{ccccc}\varphi_{i,j}(\xi)&\varphi_{i,1}(\xi)&\ldots&\varphi_{i,j}(\xi)\\ \varphi_{i,1}(\xi)&\varphi_{i,2}(\xi)&\ldots&\varphi_{i,j+1}(\xi)\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \varphi_{i,j}(\xi)&\varphi_{i,j+1}(\xi)&\ldots&\varphi_{i,2j}(\xi)\end{array}\right|

If we apply a well-known transformation formula to this latter determinant (see e.g., [11]), we deduce:

\left.\phi_{i,j}(\xi)=\frac{1}{(j+1)!}\mathrm{F}_{t_{1},t_{2},\cdots,t_{j+1}}\left[\prod_{V=1}^{j+1}\left(\xi-t_{V}\right)^{i}\right)V^{2}\left(t_{1},t_{2},\ldots,t_{j+1}\right)\right]

where we note $\mathrm{cu}\mathrm{F}_{t_{1},t_{2}},\cdots,t_{j+1}$ successive application of the operator F to the variables $t_{1},t_{2},\ldots,t_{j+1}$ .
24. - To effectively determine all Gauss-type formulas, it is sufficient to solve the system obtained from (10) if we replace Q with $n$ polynomials of degree $n-1$ linearly independent. This comes down to solving system (14) or an equivalent system in the sense of no. 6.

We will deal in particular with the case when $n>1,1\leqq u<n$ and $r_{u+1}=r_{u+2}==\ldots=r_{n}=1$ The other orders of multiplicity $r_{1},r_{2},\ldots.r_{u}$ are odd numbers (some or all of them can also be equal to 1).

Assuming we have obtained the nodes $x_{1},x_{2},\ldots,x_{u}$ , the other nodes are uniquely determined as the roots of the orthogonal polynomial $\rho_{n-u}$ very sad $n-u$ attached to functionality ⁹ )

\mathrm{C}[\mathrm{f}]=\mathrm{A}\left[\left(\prod_{i=1}^{n}\left(x_{i}-xr_{i}+1\right)f\right]\right.

polynomial $\rho_{n-u}$ can be obtained by solving a linear system. If we put

\rho_{n-n}=(-1)^{n-u}\sum_{v=0}^{n-u}d_{v}(\xi)(\xi-x)^{v}\quad\left(d_{n-n}(\xi)=1\right)

HAVE

\sum_{v=0}^{n-u}d_{v}(\xi)r_{i+v,0}(\xi)=0,\quad i=0,1\ldots,n-u-1

(43)

function $\mathrm{A}[f]$ , having the order of positivity $k\geq\frac{1}{2}(p+n+1)$ , functional $\mathrm{C}[f]$ will have the order of positivity $k-\frac{1}{2}\left(r_{1}+r_{2}+\ldots+r_{u}+u\right)\geq n-u$ and therefore the system (43) has a unique well-determined solution. The determinant of this system is independent of $\xi$ and is equal to $\Gamma_{n-u}$ .
25.- System (43) is equivalent to the last $n-u$ equations (14). Taking into account this system, the first $u$ We will replace equations (14) with others that will contain only the variables $x_{1},x_{2},\ldots,x_{u}$ .

For this we consider the functional

\mathrm{C}^{(j)}[f]=\mathrm{A}\left[\prod_{\begin{subarray}{c}i=1\\ i\neq j\end{subarray}}^{u}\left(x_{i}-x_{i}r_{i}+1\right)f\right]

(44)

Then $j^{a}$ equation (14) can be written
(45)

\mathbb{C}^{(j)}\left[\left(x_{j}-x\right)^{r}j\rho_{u-n}^{2}\right]=0

We now observe that $\mathrm{C}[f]=\mathrm{C}^{(j)}\left[\left(x_{j}-x\right)^{r}j^{+1}f\right]$ so $y_{i,0}\left(x_{j}\right)=r_{r}^{(j)}{}_{j+i+i,0}\left(x_{j}\right)$ , $i=0,1\ldots$ , so for $\xi=x_{j}$ , the system (43) becomes

⁰⁰footnotetext: 9. We can replace

\left(x-x_{i}\right)^{r}i^{+1}\mathrm{cu}\left(x_{i}-x\right)^{r}i^{+1}

because the numbers

r_{i}+1

I am a parent.

(40)

\sum_{v=0}^{n-u}d_{v}\left(x_{j}\right)r_{rj+1+i+v,0}^{(j)}\left(x_{j}\right)=0,\quad i=1,\ldots,n-u-1

If we now take this system into account, equation (45) becomes
(47)

\sum_{v=0}^{n-u}d_{v}\left(x_{j}\right)r_{rj+v,c}^{(j)}\left(x_{j}\right)=0

Eliminating the $d_{v}\left(x_{j}\right),v=0,1,\ldots,n-u-1$ from those $n-u+1$ equations (56 (47), we find
(48)

\Gamma_{r_{j}^{\prime},n-u}^{(j)}\left(x_{j}\right)=0.

If we do this $j=1,2,\ldots,u$ we find a system that determines our nodes $x_{1},x_{2},\ldots,x_{u}$ .

Based on what was said in No. 23, it can be written

\Gamma_{r_{j},n-u}^{(j)}\left(x_{j}\right)=\frac{1}{(n-u+1)!}C_{t_{1},t_{2},\ldots,t_{n-u}+1}^{(j)}\left[\prod_{\nu=1}^{n-u+1}\left(x_{j}-t_{\nu}\right)^{r_{j}}\right)

\left.V^{2}\left(t_{1},t_{2},\ldots,t_{n-u+1}\right)\right]

and if we take into account (44),

\begin{gathered}\Gamma_{r_{j},n-u}^{(j)}\left(x_{j}\right)=\\ =\frac{1}{(n-u+1)!}\mathrm{A}_{t_{1},t_{2},\ldots,t_{n-u+1}}\left[\left(\prod_{\begin{subarray}{c}i=1\\ i\neq j\end{subarray}}^{u-u=1}\prod_{v=1}^{n-u+1}\left(x_{i}-t_{v}\right)^{r_{i}+1}\right)^{n-u+1}\left(x_{v=1}^{n-u+1}t_{v}\right)^{r}j\right)\\ \left..V^{2}\left(t_{1},t_{2},\ldots,t_{n-u+1}\right)\right]\end{gathered}

Once the nodes $x_{1},x_{2},\ldots,x_{u}$ determined from the indicated system, the polynomial can be found $\rho_{n-u}$ calculating from system (47) the coefficients $d_{v}\boldsymbol{x}_{j}{}^{\prime},v=0,1,\ldots$ …, $n-u-1$ . We can write this polynomial explicitly using the moments of the functional $\mathrm{C}[f]$ We have

	we		(49)
	$\displaystyle\rho_{n-n}(x)=\frac{\Gamma_{1},n-u-1(x)}{\Gamma_{n-u-1}}$

If $\mathrm{A}[f]$ has a positive order $\geqq r_{1}+r_{2}+\ldots+r_{u}+n$ , we can obtain the polynomial $\rho_{n-u}$ and with the help of a well-known formula of Christoffel (see e.g. [11]). For this let us denote with $\mathrm{P}_{m}(x)$ the orthogonal polynomial of degree $m$ attached to the function $\mathrm{A}[f]$ This polynomial is well determined for $m\leq n+r_{1}+r_{2}+\ldots+r_{u}$ .

Then the polynomial $\rho_{n-u}(x)\prod_{i=1}^{u}\left(x-x_{i}\right)^{r}{}^{r+1}$ differs only by ^a constant factor of

StCeroStlasi,VI-1

\mathrm{V}\binom{\mathrm{P}_{n-u},\mathrm{P}_{n-u+1},\ldots.,\mathrm{P}_{n+r_{1}+r_{2}+\ldots+r_{u}}}{\underbrace{x_{1},x_{1},\ldots.,x_{1}}_{r_{1}+1},\underbrace{x_{2},x_{2},\ldots,x_{2}}_{r_{2}+1},\ldots,\underbrace{x_{u},x_{u},\ldots,x_{u},x}_{r_{u}+1}}

26.

In particular, if $u=1$ , functional $C^{(1)}[f]$ it reduces to $A[f]$ and it is deduced that the node $x_{1}$ is a root of the polynomial

	$\displaystyle A_{r_{1},n-1}(x)=$	$\displaystyle\left\|\begin{array}[]{llll}\alpha_{r_{1},0}(x)&\alpha_{r_{1},1}(x)&\ldots&\alpha_{r_{1},n-1}(x)\\ \alpha_{r_{1},1}&(x)&\alpha_{r_{1},2}(x)&\ldots\\ \cdot&\cdot&\alpha_{r_{1},n}(x)\\ \cdot&\cdot&\cdot&\cdot\\ \alpha_{r_{1},n-1}(x)&\alpha_{r_{1},n}(x)&\ldots&\alpha_{r_{1},2n-2}(x)\end{array}\right\|=$
	$\displaystyle\frac{1}{n!}A_{t_{1},t_{2},\ldots,t_{n}}\left[\left(\prod_{\nu=1}^{n}\left(x-t_{\nu}\right)\right)^{\left.r_{1}1V^{2}\left(t_{1},t_{2},\ldots,t_{n}\right)\right]}\right.$	$\displaystyle=$

which, if $r_{1}=1$ , differs only by a constant factor from the orthogonal polynomial of degree $n$ attached to the functionality $A[f]$ .

Regarding the calculation of the polynomial $\rho_{n-1}$ , in this case, we can apply formula (49). If $\AA [f]$ has a positivity order $\geq n+r_{1}$ HAVE

	$\displaystyle\mathrm{V}(\underbrace{\begin{array}[]{l}\mathrm{P}_{n-1},\mathrm{P}_{n},\ldots,\mathrm{P}_{n+r_{1}-1}\\ x_{1},x_{1},\ldots,x_{1}\end{array}}_{r_{1}+1})\rho_{n-1}(x)\left(x-x_{1}\right)_{r_{1}+1}=$		(50)
	$\displaystyle=\mathrm{V}\left(\begin{array}[]{l}\mathrm{P}_{n-1},\mathrm{P}_{n},\ldots,\mathrm{P}_{n+r_{1}}\\ x_{1},x_{1},\ldots,x_{1},x\\ r_{1}+1\end{array}\right)\cdot$

From the determinant in the second member of formula (50) one can easily extract the factor $\left(x-x_{1}\right)r_{1}+1$ because, first of all, the element $\mathrm{P}_{n+i}(x)$ can be replaced, based on the others $r_{1}+1$ align with ( $i=-1,0,1,2,\ldots,r_{1}$ )
$\sum_{v=r_{1}+1}^{n+i}\frac{\left(x-x_{1}\right)^{v}}{v!}\mathrm{P}_{n+i}^{v}\left(x_{1}\right)=\left(x-x_{1}\right)^{r_{1}+1}\sum_{v=r_{1}+1}^{n+i}\frac{\left(w-x_{1}\right)^{v}-r_{1}-1}{v!}P_{n+i}^{(v)}\left(x_{1}\right)$
27. We can calculate on $R\left[x^{p+n+1}\right]$ relative to the rest of the Gauss-type formulas thus obtained. From formulas (36), (44) we deduce

\mathrm{R}\left[x^{p+n+1}\right]=\mathrm{C}^{(j)}\left[\left(x_{j}-x\right)^{r}j+1\rho_{n-u}^{2}\right]

and if we take into account (46) we obtain

\left.\mathrm{R}\mid x^{p+n+1}\right]=\sum_{\nu=0}^{n-u}d_{\nu}\left(x_{j}\right)\gamma_{r^{\prime}j+1+n-u+\nu,0}^{(i)}\left(x_{j}\right)

(51)

(51), we obtain
(52)

\mathrm{R}\left[x^{p+n+1}\right]=\frac{\Gamma_{r_{j}^{\prime}+1,n-u}^{(j)}\left(x_{j}\right)}{\Gamma_{r_{j}+1,n-u-1}^{(j)}\left(x_{j}\right)}

Of course, in this formula we need to replace the nodes $x_{1},x_{2},\ldots$ , $x_{u}$ with their values calculated from the system that is deduced from (48) if we do $j=1,2,\ldots,u$ .

In particular if $u=1$ , we have

\mathrm{R}\left[x^{p+n+1_{1}}\right]=\frac{Ar_{1+1,n-1}\left(x_{1}\right)}{Ar_{1+1,n-2}\left(x_{1}\right)}

In general, the calculation of (51) is complicated. Only if $r_{1}=r_{2}=\ldots r_{u}=1$ (the Gaussian formula is then unique) we have the well-known value fold (in this case $p=n-1$ )

\mathrm{R}\left[x^{2n}\right]=\frac{A_{n}}{A_{n-1}}

28.

Let's consider the particular case $n=2,r_{1}=3,r_{2}=1$ and let's assume that $\mathrm{A}[f]$ has a positivity order $\geqq 3$ . The node $x_{1}$ is a root of the equation

\left(\alpha_{0}\alpha_{2}-\alpha_{1}^{2}\right)x^{6}+3\left(\alpha_{1}\alpha_{2}-\alpha_{0}\alpha_{3}\right)x^{5}+3\left(\alpha_{1}\alpha_{3}+\alpha_{0}\alpha_{4}-2\alpha_{2}^{2}\right)x^{4}+\left(8\alpha_{2}\alpha_{3}-\right.

(53)

$\left.-7\alpha_{1}\alpha_{4}-\alpha_{0}\alpha_{5}\right)x^{3}+3\left(\alpha_{2}\alpha_{4}+\alpha_{1}\alpha_{5}-2\alpha_{3}^{2}\right)x^{2}+3\left(\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}\right)x+\alpha_{3}\alpha_{5}-\alpha_{+}^{2}=0$
Node $x_{2}$ is given by the formula

x_{2}=\frac{\alpha_{1}x_{1}^{3}-3\alpha_{2}x_{1}^{2}+3\alpha_{3}x_{1}-\alpha_{4}}{\alpha_{0}x_{1}^{3}-3\alpha_{1}x_{1}^{2}+3\alpha_{2}x_{1}-\alpha_{3}}

and the number $\mathrm{R}\left[x^{6}\right]$ of

		$\displaystyle R\left[x^{6}\right]=\alpha_{2}x_{1}^{4}-4\alpha_{3}x_{1}^{3}+6\alpha_{4}x_{1}^{2}-4\alpha_{5}x_{1}+\alpha_{6}-$
		$\displaystyle\quad-\frac{\left(\alpha_{1}x_{1}^{4}-4\alpha_{2}x_{1}^{3}+6\alpha_{3}x_{1}^{2}-4\alpha_{4}x_{1}+\alpha_{5}\right)^{2}}{\alpha_{0}x_{1}^{4}-4\alpha_{1}x_{1}^{3}+6\alpha_{2}x_{1}^{2}-4\alpha_{3}x_{1}+\alpha_{4}}$

Equation (53) has at least two real roots, because, based on the assumptions made, it is of even degree and has at least one real root ^10. We therefore have at least two Gauss-type formulas, of which, however, in general, only one corresponds to the minimizing system.

To show this, it is enough to conveniently customize the moments $\alpha_{0}=1,\ \alpha_{1}=0,\ \alpha_{2}=4,\ \alpha_{3}=24,\ \alpha_{4}=200,\ \alpha_{5}=-408$ .

Then equation (53) reduces to

(x^{2}+2x-8)(x^{4}-20x^{3}+174x^{2}-214x+1556)=0,

which has only two real roots, $2$ and $-4$ These are the node values $x_{1}$ in the two corresponding Gauss-type formulas.

Node values $x_{2}$ are $-13$ , $5$ , and its values $R[x^{6}]$ are $-12920+\alpha_{6}$ , respectively $-10760+\alpha_{6}$ .

The two Gaussian formulas can be written

		$\displaystyle A[f]\approx\frac{1}{3375}\left[\left(3367f(2)-6630f^{\prime}(2)+12600f^{\prime\prime}(2)+8f(-13)\right]\right.$
		$\displaystyle A[f]\approx\frac{1}{729}\left[593f(-4)+1696f^{\prime}(-4)+1782f^{\prime\prime}(-4)+136f(5)\right]$

The first formula alone corresponds to a minimizing system.
29. We will say that the functional $\mathrm{A}[f]$ is symmetric of the order $k$ if by a linear transformation of the variable $x$ we can make moments with odd indices $\alpha_{2i-1},i=12,\ldots,k$ to become null. Such functionals are, for example, those of the form $\int^{j}\rho(x)f(x)dx$ , where $a,b$ are finite and the function $\rho(x)$ check the property $\rho(x)=\rho(b+a-x)$ Also functions of the form $\int_{-\infty}^{\infty}\rho(x)f(x)dx$ , where $\rho(x)$ is a para function.

Returning to the case $n=2$ . $r_{1}=3,r_{2}=1$ studied above, let's assume that the functional $\mathrm{A}[f]$ has a positivity order $\geq 3$ and $\alpha_{5}=0$ yes an order of symmetry $\geqq 3$ We can then assume $\alpha_{1}=\alpha_{3}=\alpha_{5}=0$ and equation (53) becomes

\alpha_{0}\alpha_{2}x^{6}+3\left(\alpha_{0}\alpha_{4}-2\alpha_{2}^{2}\right)x^{4}+3\alpha_{2}\alpha_{4}x^{2}-\alpha_{4}^{2}=0

(54)

In the conditions we are in ( $\alpha_{0}>0,\alpha_{2}>0,\alpha_{0}\alpha_{4}-\alpha_{2}^{2}>0$ ) it is immediately seen that this equation has only two real roots unequal and equal in absolute value ¹¹ ). We therefore have two Gauss-type formulas with the same $\mathrm{R}\left[x^{6}\right]$ .

In particular, for functionality $J^{(0,0)}[f]$ , we have $\alpha_{0}=2,\alpha_{2}=\frac{2}{3}$ , $\alpha_{4}=\frac{2}{5}$ sj the roots of equation (54) are $\frac{1}{\sqrt{5}},-\frac{1}{\sqrt{5}}$ . Doing the calculations, we obtain the Gauss-type formula

\begin{gathered}f^{(0,0)}[f]=\frac{1}{128}\left[175f\left(\frac{1}{\sqrt{5}}\right)-\frac{40}{\sqrt{5}}f^{\prime}\left(\frac{1}{\sqrt{5}}\right)+\frac{32}{3}f^{\prime\prime}\left(\frac{1}{\sqrt{5}}\right)+81f\left(-\frac{\sqrt{5}}{3}\right)\right]+\\ +\frac{128}{1575}\mathrm{D}_{6}[f]\end{gathered}

and a second Gauss-type formula, with the same remainder, which is deduced from this by replacing $\sqrt{5}\mathrm{cu}-\sqrt{5}$ .
30. We will also consider the case when $n=3,r_{1}=3,r_{2}=r_{3}=1$ , assuming that $A[f]$ has a positivity order $\geq 4$ and an order of symmetry $\geq 4$ Then we can assume $\alpha_{1}=\alpha_{3}=\alpha_{5}=\alpha_{7}=0$ and the knot $x_{1}$ is 0 root of the equation

⁰⁰footnotetext: 11. For the discussion it is enough to assume

\alpha_{0}=\alpha_{2}=1

. Then if

\alpha_{4}\geq 2

, the property results from Descartes' rule of signs and if

1<\alpha_{4}<2

from the fact that the derivative of the equation in

x^{2}

it has no real roots.

(55)

		$\displaystyle\alpha_{2}\left(\alpha_{0}\alpha_{4}-\alpha_{2}^{2}\right)x^{9}+3\left(\alpha_{2}^{2}\alpha_{4}-2\alpha_{0}\alpha_{4}^{2}+\alpha_{0}\alpha_{2}\alpha_{6}\right)x^{7}+3\left(3\alpha_{2}\alpha_{4}^{2}-4\alpha_{2}^{2}\alpha_{6}+\right.$
		$\displaystyle\left.+\alpha_{0}\alpha_{4}\alpha_{6}\right)x^{5}+\left(11\alpha_{2}\alpha_{4}\alpha_{6}-10\alpha_{4}^{3}-\alpha_{0}\alpha_{6}^{2}\right)x^{3}+3\alpha_{6}\left(\alpha_{4}^{2}-\alpha_{2}\alpha_{6}\right)x=0.$

Because $\alpha_{2}(\alpha_{0}\alpha_{4}-\alpha_{2}^{2})>0$ , $\alpha_{6}(\alpha_{4}^{2}-\alpha_{2}\alpha_{6})<0$ , this equation has at least 3 real roots, namely the root $0$ and two other different and equal in absolute value ^12. So we have at least three Gauss-type formulas.

The other two nodes are the roots of the orthogonal polynomial of degree $2$ attached to the functional $C[f]$ appropriate.

Apart from a constant factor different from zero, this polynomial, based on formula (49), can be written in the form

\begin{gathered}{\left[\left(\alpha_{0}x_{1}^{4}+6\alpha_{2}x_{1}^{2}+\alpha_{4}\right)x+4\left(\alpha_{2}x_{1}^{3}+\alpha_{4}x_{1}\right)\right]\left[\left(\alpha_{2}x_{1}^{4}+6\alpha_{4}x_{i}^{3}+\alpha_{6}\right)x+4\left(\alpha_{4}x_{1}^{9}+\right.\right.}\\ \left.\left.+\alpha_{6}x_{1}\right)\right]-\left[4\left(\alpha_{2}x_{1}^{3}+\alpha_{4}x_{1}\right)x+\alpha_{2}x_{1}^{4}+6\alpha_{4}x_{1}^{2}+\alpha_{6}\right]^{2}\end{gathered}

number $R\left[x^{8}\right]$ relative to the rest of the formula is equal to

$\frac{\left|\begin{array}[]{ccr}\alpha_{0}x_{1}^{4}+6\alpha_{2}x_{1}^{2}+\alpha_{4}&4\alpha_{2}x_{1}^{3}+4\alpha_{4}x_{1}&\alpha_{2}x_{1}^{4}+6\alpha_{4}x_{1}^{2}+\alpha_{6}\\ 4\alpha_{2}x_{1}^{3}+4\alpha_{4}x_{1}&\alpha_{2}x_{1}^{4}+6\alpha_{4}x_{1}^{2}+\alpha_{6}&4\alpha_{4}x_{1}^{3}+4\alpha_{6}x_{1}\\ \alpha_{2}x_{1}^{4}+6\alpha_{4}x_{1}^{2}+\alpha_{6}&4\alpha_{4}x_{1}^{3}+4\alpha_{6}x_{1}&\alpha_{4}x_{1}^{4}+6\alpha_{6}x_{1}^{2}+\alpha_{8}\end{array}\right|}{\left|\begin{array}[]{cc}\alpha_{0}x_{1}^{4}+6\alpha_{2}x_{1}^{2}+\alpha_{4}&4\alpha_{2}x_{1}^{3}+4\alpha_{4}x_{1}\\ 4\alpha_{2}x_{1}^{3}+4\alpha_{4}x_{1}&\alpha_{2}x_{1}^{4}+6\alpha_{4}x_{1}^{2}+\alpha_{6}\end{array}\right|}$

If the node $x_{1}$ is equal to 0, the other two nodes are
$\sqrt{\frac{\alpha_{6}}{\alpha_{4}}},-\sqrt{\frac{\alpha_{6}}{\alpha_{4}}}$ , and $\mathrm{R}\left[x^{8}\right]=\frac{\alpha_{4}\alpha_{8}-\alpha_{6}^{2}}{\alpha_{4}}$ so we find the Gaussian formula $\AA [f]\approx\frac{1}{2\alpha_{6}^{2}}\left[2\left(\alpha_{0}\alpha_{6}^{2}-\alpha_{4}^{3}\right)f(0)+\left(\alpha_{2}\alpha_{6}-\alpha_{4}^{2}\right)\alpha_{0}f^{\prime\prime}(0)+\alpha_{4}^{3}\left[f\left(\sqrt{\frac{\alpha_{6}}{\alpha_{4}}}\right)+f\left(-\sqrt{\frac{\alpha_{6}}{\alpha_{4}}}\right)\right]\right]$

In particular we have the formulas

\begin{gathered}\begin{aligned} J^{(\alpha,\alpha)}[f]=&\frac{2^{2\alpha+2}\Gamma(\alpha+1)\Gamma(\alpha+3)}{25\Gamma(2\alpha+0)}\left[8\left(\alpha+1,\left(2\alpha+57,f(0)+20(\alpha+1)f^{\prime\prime}(0)+\right.\right.\right.\\ &\left.+3(2\alpha+7)^{2}\left(f\left(\sqrt{\frac{5}{2\alpha+7}}\right)+f\left(-\sqrt{\frac{5}{2\alpha+7}}\right)\right)\right]+\\ &\left.+15\frac{2^{2\alpha+7}\Gamma(\alpha+1)\Gamma(\alpha+5)}{(2\alpha+7)\Gamma(2\alpha+10)}D_{8}[f]\quad(\alpha>-1)^{13}\right)\end{aligned}\\ \begin{aligned} I^{(0,0)}[f]=&\frac{1}{375}\left[456f(0)+20f^{\prime\prime}(0)+147\left(f\left(\sqrt{\frac{5}{7}}\right)+f\left(-\sqrt{\frac{5}{7}}\right)\right)\right]+\\ &+\frac{8}{441}D_{8}[f]\end{aligned}\end{gathered}

12.

It would also be interesting here to demonstrate that, under the conditions of the problem, equation (55) has only three real roots.
13.

In this formula $\Gamma(x)$ is the second-order Eulerian function.
$\mathrm{H}\mid f]=\frac{\sqrt{\pi}}{50}\left[44f(0)+5f^{\prime\prime}(0)+3\left(f\left(\sqrt{\frac{5}{2}}\right)+f\left(-\sqrt{\frac{5}{2}}\right)\right)\right]+\frac{15\sqrt{\pi}}{8}\mathrm{D}_{8}[f]$ .

31.

To show that not all three formulas correspond to a minimizing system in the sense of $\mathdollar 2$ , let us consider the particular case when $\alpha_{0}=1,\alpha_{2}=2,\alpha_{4}=8,\alpha_{6}=\frac{288}{5}$ Equation (55) then becomes

x\left(x^{2}-4\right)\left(25x^{6}+100x^{4}+400x^{2}+6912\right)=0

which has real roots 0,2 and - 2. These being the possible values of the node $x_{1}$ , the other two nodes $x_{2},x_{3}$ are respectively the roots of the equations $5x^{2}-36=0,5x^{2}+20x+16=0$ , and $5x^{2}-20x+16=0$ The corresponding values of $\mathrm{R}\left[x^{8}\right]$ SYNTHESIS $\alpha_{8}-\frac{2^{7}.81}{5^{2}}$ for $x=0$ and $\alpha_{8}-\frac{2^{7}.89}{5^{2}}$ for

		The Gauss-type formulas thus obtained are
		$\displaystyle\qquad\mathrm{A}[f]\approx\frac{1}{3\overline{24}}\left[274f(0)+144f^{\prime\prime}(0)+25\left(f\left(\frac{6}{\sqrt{5}}\right)+f\left(-\frac{6}{\sqrt{5}}\right)\right)\right]$
		$\displaystyle+\frac{25}{2}(91+07\sqrt{5})f\left(\frac{1}{2.19^{8}}\left[6443f(2)-3686f^{\prime}(2)+1444f^{\prime\prime}(2)+\right.\right.$
		$\displaystyle\mathrm{A}[f]\approx\frac{1}{2.19^{3}}\left[6443f(-2)+3686f^{\prime}(-2)+1444f^{\prime\prime}(-2)+\right.$
		$\displaystyle+\frac{25}{2}\left(291-107\sqrt{5}f\left(\frac{10+2\sqrt{5}}{5}\right)+\frac{25}{2}(291+107\sqrt{5})f\left(\frac{10-2\sqrt{5}}{5}\right)\right]$

32.

For the case when all multiplicity orders are equal to each other, we have the following property due to P. Turán [12].

Theorem 7. For any linear functional $\mathrm{A}[f]$ which has the order of positivity $k$ , relative to any natural number n and to any system of $n$ multiplicity orders, all equal to the same odd number $r$ so that $k\geqq\frac{1}{2}n(r+1)$ , there is one and only one Gaussian formula (8).

P. Turá's proof consists in observing that if, in this case, the orthogonality condition (10) is verified, the polynomial $\pi=\left(x-x_{1}\right)\left(x-x_{2}\right)\ldots\left(x-x_{n}\right)$ is minimizing. Indeed, either $\mathrm{Q}\epsilon P_{n}$ a polynomial different from $\pi$ We then have

\mathrm{Q}^{r+1}-\pi^{r+1}=(r+1)(\mathrm{Q}-\pi)\pi^{r}+\frac{r+1}{2}\left[(\mathrm{Q}-\pi)\pi^{\frac{r-1}{2}}\right]^{2}+

+\sum_{\nu=1}^{r-1}\nu\left[\left(\mathrm{Q}^{2}-\pi^{2}\right)\mathrm{Q}^{\frac{r-3}{2}-\nu+1}\pi^{\nu-1}\right]^{2}

and finally taking into account orthogonality, the fact that $Q-\pi$ is a polynomial of degree $n-1$ , and

(\mathrm{Q}-\pi)^{\frac{r-1}{2}},\left(\mathrm{Q}^{2}-\pi^{2}\right)\mathrm{Q}^{\frac{r-3}{2}-\nu+1}\pi^{\nu-1},\nu=1.2,\ldots,\frac{r-1}{2}

are polynomials of degree $\frac{1}{2}n(r+1)-1\leqq k-1$ , we deduce

\begin{gathered}\mathrm{A}\left[\mathrm{Q}^{r+1}\right]-\mathrm{A}\left[\pi^{r+1}\right]=\frac{r+1}{2}\mathrm{~A}\left[\left((\mathrm{Q}-\pi)\pi\frac{r-1}{2}\right)^{2}\right]+\\ +\sum_{\nu=1}^{\frac{r-1}{2}}\nu\mathrm{~A}\left[\left(\left(\mathrm{Q}^{2}-\pi^{2}\right)\mathrm{Q}^{\frac{r-3}{2}-\nu+1}\pi^{-1}\right)^{2}\right]\\ \geq\frac{r+1}{2}\mathrm{~A}\left[\left((\mathrm{Q}-\pi)\pi^{\frac{r-1}{2}}\right)^{2}\right]>0\end{gathered}

so $\mathrm{A}\left[Q^{r+1}\right]>\mathrm{A}\left[\pi^{r+1}\right]$ , which proves the theorem. It is seen that the proof remains valid for $r=1$ .

BIBLIOGRAPHY

1.

Cauchy A., Sur les fonctions Interpolaires. Comptes rendus Ac. Sci. Parls, 11, pp. 775,789, 1840.
2.

Jackson D., The theory of approximation, 1930.
3.

Ditto thons. Year hons. A. Diff Ma1cm., (2), 25, pp. 184,152, 1924.
4.

Mar a G. Sur, Dalgorithme toulour,
5.

Po1 ya G., Jur un algorithme toujours convergent pourobtenir les polynomes de mention continuous, quel coque. Fully rendered Ac. Sci. Parls, 157, pp, 840-843, 1913.
6.

Popové timatic, Burresti, 1926, pr 35.38, algebraic, Bul. Soc. Studentilor in ma*
7.

Ditto Sur a is bem Gis.
8.

Idem Acad. Roumaine, XVI, pp. 214=217, 1934.
9.

Id Work on the formal remainder in some approximation formulas of analysis, sess. $t., Acad RPR, 1950, pp. 183:186.
10.

Ditto Circle. Math., III, pp. 53:122, 1952.

On a generalization of the Gauss formula for numerical integration

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