1. 1.

   The remainder can be expressed in several ways.$R[f]$of the quadrature formula

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Or$x_{\alpha}(\alpha=1,2,\ldots,n)$are distinct points of the real axis and$A_{\alpha},\alpha=1,2,\ldots,n$given real constants, independent of the function$f$. We will assume that$has$And$b$, Or$a<b$, are finite, that all the functions considered are real, and we will denote by$I$an interval containing the points$a,b,x_{\alpha},(\alpha=1,2,\ldots,n)$.

Either$m$the degree of accuracy of the remainder$R[f]$(or the quadrature formula (1)), therefore the number, well determined by the conditions:$R[1]=R[x]=\cdots=R\left[x^{m}\right]=0$,$R\left[x^{m+1}\right]\neq 0$. If$R[1]\neq 0$we take$m=-1$And$\operatorname{si}R[1]=0$We have$0\leqq m\leqq 2n-1$.

If$f$is a continuous function on the interval$I$we have [1]

The points$\xi_{\alpha}$, on the one hand, and the points$\xi_{\alpha}^{\prime}$, on the other hand, are distinct but generally depend on the function$f$. The constants$A,B$are independent of the function$f$. We have$A+B=R\left[x^{m+1}\right]$And$\left[y_{1},y_{2},\ldots,y_{r};f\right]$denotes the divided difference, of order$r-1$, of the function$f$on the points (or nodes)$y_{1},y_{2},\ldots,y_{r}$. If$m\geqq 0$the points$\xi_{\alpha},\xi_{\alpha}^{\prime}$, can be chosen within the interval$I$and if in addition the function$f$has a ($m+1$)-th derivative$f^{(m+1)}$inside$I$, we have

$\xi,\xi^{\prime}$, being two points inside the interval$I$And$A,B$being, moreover, the same constants as in (2).

If we can take$B=0$in (2) or in (3), we say that the remainder$R[f]$is of the simple form. We have previously given necessary and sufficient conditions for this to be so [1], but the rest is not always of the simple form.

We shall show that even if the remainder is not of the simple form, in the preceding sense, we can, in certain cases, introduce
modified divided differences so that the remainder becomes of the simple form with respect to these new divided differences. The definition of the divided difference which will occur here will result from the following.
2. Let always$m$the degree of accuracy of$R[f]$and designate by$k(0\leqq k\leqq n)$the number of points$x_{\alpha}$included in the open interval ($a,b$). If$k>0$we can assume that the points$x_{1},x_{2},\ldots,x_{k}$, are in ($a,b$) and if$k<n$that the points$x_{k+1}$,$x_{k+2},\ldots,x_{n}$, are outside of ($a,b$). Let us designate by$c,d(c\leqq a<b\leqq d)$, the ends of the interval$I$and consider the polynomials

which are well defined since$R\left[(x-c)^{m+1}\right]=R\left[x^{m+1}\right]\neq 0$and which verify the equalities

It is easy to see that we must have$m\leqq n+k-1$. Indeed if we take the polynomial

(whose form is easy to write if$k=0$Or$k=n$), We have

Let us now designate by$D\left(y_{1},y_{2},\ldots,y_{n+k+1};f\right)$the determinant of function values$P_{0},P_{1},\ldots,P_{m},P_{m+2},P_{m+3},\ldots,P_{n+k},f$(if$m=-1$functions$P_{1},P_{2},\ldots$,$\left.P_{n+k},f\right)$on the points$y_{1},y_{2},\ldots,y_{n+k+1}$, and by$D^{*}\left(y_{1},y_{2},\ldots,y_{n+k}\right)$the minor corresponding to the element$f\left(y_{n+k+1}\right)$of this determinant, therefore the determinant of the values ​​of the functions$P_{0},P_{1},\ldots,P_{m},P_{m+2},P_{m+3},\ldots,P_{n+k}$on the points$y_{1},y_{2},\ldots,y_{n+k}$. We have$D\left(y_{1},y_{2},\ldots,y_{n+k+1};x^{m+1}\right)=(-1)^{n+k-m-1}V\left(y_{1},y_{2},\ldots,y_{n+k+1}\right)$Or$V\left(y_{1},y_{2},\ldots\right.$,$y_{n+k+1}$) is the Vandermonde determinant of numbers$y_{1},y_{2},\ldots,y_{n+k+1}$.

The new divided difference we introduce is defined by the formula

In this definition we have assumed that the points$y_{\alpha}$are distinct. But we can extend the definition also to the case of points$y_{\alpha}$not necessarily distinct by a passage to the limit in formula (6). This amounts to maintaining the definition formula (6) by suitably modifying the determinant$D$(and also the determinants
$\left.D^{*},V\right)$as we explained previously [1]. This requires the existence of a certain number of derivatives of the function$f$. In this way the divided difference (6) is defined whatever the points$y_{\alpha}$distinct or not.
3. With the previous notations consider the continuous function$\varphi(x)$which outside the points$x_{\alpha}$is equal to the linear combination$l(x)\left[x_{1},x_{1},x_{2},x_{2},\ldots\right.$,$\left.x_{k},x_{k},x_{k+1},x_{k+2},\ldots,x_{n},x;f\right]^{*}$functions$P_{0},P_{1}\ldots,P_{m},P_{m+1},P_{m+2},\ldots,P_{n+k},f$. We then have$\varphi\left(x_{\alpha}\right)=0(\alpha=1,2,\ldots,n)$so also

and taking into account (4), (5), we have

Noting that the function$l$does not change sign on$(a,b)$, it follows that

where the$\xi_{\alpha}$are distinct points of the interior of$I$(generally depending on the function$f$), provided that the mean theorems can be applied to divided differences (6). I refer the reader for these theorems to a previous work [1]. This condition is surely verified if:
(H). The determinant$D^{*}\left(y_{1},y_{2},\ldots,y_{n+k}\right)$East$\neq 0$, whatever the points$y_{1}$,$y_{2},\ldots,y_{n+k}$distinct or not.

In the previous demonstration of formula (7) we assumed that the function continues$f$has a derivative (at least on the points$x_{1},x_{2},\ldots,x_{k}$). We can easily see that the result obtained is valid under the sole hypothesis of continuity on the closed interval$[a,b]$of the function$f$.

If we designate by$W\left(g_{1},g_{2},\ldots,g_{s}\right)$the Wronskian of functions$g_{1},g_{2},\ldots,g_{s}$, if we further assume that the function$f$have a$(n+k)$-th derivative on the interior of$I$and if the condition$(H)$is verified, we have (the Wronskian has its rows and columns in the usual order)

$\xi$being an interior point of$I$. Note that the second member of this formula
is of the form

Or$Q_{\alpha}$is a polynomial of degree$\alpha$independent of function$f(\alpha=0,1,\ldots$,$n+k-m-1$).
4. If$p_{0}=1$And$p_{\alpha}(\alpha=0,1,\ldots,n+k)$are the fundamental symmetric functions of numbers$y_{\alpha}-c,(\alpha=1,2,\ldots,n+k)$, We have

It follows that the condition ($H$) is checked if the numbers

are all positive or all negative.
Example. Consider the quadrature formula

In this case we have$k=m=1$and we can take$c=0,d=2$. The rest is not of the usual simple form (we have$R[f]=\frac{7}{375}f^{\prime\prime}(\xi)-\frac{32}{375}f^{\prime\prime}\left(\xi^{\prime}\right)$if the second derivative exists). In this case the numbers (8) are all of the same sign and different from 0, so the condition$(H)$is verified. If the function$f$is continuous on$[0,2]$and has a fourth derivative on$(0,2)$, We have

1. 5.

   The above considerations can easily be extended to quadrature formulas where the second member also linearly contains some of the values ​​on the points$x_{\alpha}$successive derivatives of the function$f$and also to linear approximation formulas corresponding to more general linear functionals.

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[1] Popoviciu, T., On the remainder in some linear approximation formulas of analysis, Mathematica, Cluj$I$(24), 95-142 (1959).

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