Theorem 1. For the linear functional$R[f]$degree of accuracy$m$, or in its simple form, it is necessary and sufficient that one has$R[f]\neq 0$, for any function$f(x)\in S$convex of order m.

For the concept and properties of convex functions (non-concave, non-convex, concave) of order$m$And for the proof of Theorem 1, the reader can consult my previous work. The function$f(x)$is said to be convex of order$m$on$I$if all its differences divided by order$m+1$, on distinct nodes, are positive. In particular in my memoirs of "Mathematica" [7, 8] one can find various applications and various generalizations of the notion of simplicity of a linear functional.

If$m\geqq 0$one can even assert that the points$\xi_{\alpha},\alpha=1,2,\ldots,m+2$of formula (3) are within the interval$I$.

If$m\geqq 0$and if$f(x)$has a derivative of order$m+1$inside$I$, we have, thanks to a formula for the important mean of A. Cauchy [1],

assuming that$R[f]$is the degree of accuracy$m$and that it is of simple form,$\xi$being a point inside the interval$I$.

Formula (4) can, in particular, be used to give an upper bound of$R[f]$when you know$f^{(m+1)}(x)$the derivative$(m+1)th$of the function$f(x)$4.
Let us return to the quadrature formula (1). We will first demonstrate that, under hypothesis H.1 and assuming that$R_{n}[f]$either degree of accuracy$m$the coefficients$d_{\alpha},\alpha=0,1,\ldots,m-1$are determined independently of$n$.

Let's calculate$R_{n}\left[x^{k}\right]$Using the well-known theory of numbers$B_{\alpha}$and
polynomials$B_{\alpha}(x)$of Bernoulli, as set forth in the classical treatise of NE Nörlund [6], we have, for$k$entire$\geqq 0$,

If we ask

($s_{0}=d_{0}+d_{1}+\ldots+d_{m-1}$), We have

By comparing formulas (5), (7) it follows that, if we set

We have$R_{n}\left[x^{k}\right]=0,k=0,1,\ldots,m$The accuracy of the last equality ($R_{n}\left[x^{m}\right]=0$) is ensured by hypothesis H. 1 (the oddness of$m$).

The system (8) determines completely and independently$n$the coefficients$d_{\alpha},\alpha=0,1,\ldots,m-1$The fact that the rest$R_{n}[f]$is indeed a degree of accuracy$m$will result from the following.

Theorem 2. If$m+n>1$and if the coefficients$d_{\alpha},\alpha=0,1,\ldots,m-1$are determined by equations (8), under the hypotheses H.1, H.2, the remainder$R_{n}[f]$is of degree of accuracy m and it is of simple form, that is to say that

Or$\xi_{\alpha},\alpha=1,2,\ldots,m+2$are$m+2$distinct points within the interval$I$(and generally depend on the function)$f(x)$).

The condition$m+n>1$is essential. Indeed, if$m+n=1$we necessarily$m=1,n=0$and then$R_{0}[f]=0$, whatever the function$f(x)$.

The demonstration now proceeds in stages by successively proving the following lemmas.

Lemma 1. If$f(x)$is a convex function of order$m$we have$R_{n}[f]-R_{n-1}[f]<0$,$n==1,2,\ldots$.

This is a consequence of Steffensen's simplicity criterion [8]. It can also be easily obtained in the following way. The difference$R[f]=R_{n}[f]$–$R_{n-1}[f]$is the remainder of the quadrature formula ($n>0$)

which is the degree of accuracy$\geqq m$This is then necessarily the formula for Cotes in the interval$[n-1,n]$relative to the nodes$n-\alpha,\alpha=0,1,\ldots,m$So we have

where we designate by$L\left(y_{1},y_{2},\ldots,y_{r};f\mid x\right)$the Lagrange interpolation polynomial of the function$f(x)$on the nodes$y_{1},y_{2},\ldots,y_{r}$We know that (for$x$different from a knot),

and the lemma follows from the fact that the polynomial$\prod_{\alpha=0}^{m}(x-n+\alpha)$is negative on the open interval$]n-1,n[$and that the second factor of the right-hand side, the difference divided by order$m+1$is, by hypothesis, positive.

In particular if$m=1$and if$f(x)$is a convex function of order 1, we have$R_{1}[f]<0$.

Lemma 2. If$m>1$and if$f(x)$is a convex function of order$m$we have

In this case, formula (1) is the Cotes formula relating to$m$knots$0,1,\ldots,m-1$The property then results from the simplicity of the rest of this formula [8].

Lemma 3. If$m>1$and if$f(x)$is a convex function of order$m$we have

The proof is still based on Steffensen's criterion, which in fact stems from the important results of J.F. Steffensen [9] on the remaining formulas of the Cotes type. Following J.F. Steffensen's exposition, we can prove Lemma 3 by first noting that we can write

Or$A[f]$is the remainder in the formula for Cotes in the interval$[m-3,m-2]$And$B[f]$the remainder in the formula of Cotes in the interval [$0,m-3$], both on the nodes$-1,0,1,\ldots,m-1$.

SO$A[f]$is the integral of$m-3$has$m-2$of the difference (for$x$(different from a knot)

and the polynomial$\prod_{\alpha=0}^{m}(x+1-\alpha)$is positive on the interval$]m-3,m-2[$It follows that if$f(x)$is convex of order$m$we have

When$m=3$we have$B[f]=0$, regardless of$f(x)$.
If$m>3,B[f]$is the integral from 0 to$m-3$of the same difference (11). Following a line of reasoning by JF Steffensen [9], we now note that the difference (11) can be written (for$x$(different from a knot)

and it follows that$B[f]$is the remainder of the formula for Cotes in the interval$[0,m-3]$on the nodes$-1,0,1,\ldots,m-2$We deduce from the considerations made by JF Steffensen [9] on the polynomial$P(x)=\prod_{\alpha=0}^{m-1}(x+1-\alpha)$that the polynomial$\int_{0}^{x}P(t)dt$is negative on the open interval$]0,m-3[$and is useless for$x=m-3$We can deduce that if$f(x)$is a convex function of order$m$we have

Formulas (10), (12), and (13) prove Lemma 3.
Theorem 2 is now easily obtained. We can conclude from the formula

Or$n\geqq m$and lemmas 1 and 2 that

for any function$f(x)$convex of order$m$.
If$m>1$it comes from the formula

Or$n\leqq m-3$and lemmas 1,3 that

for any function$f(x)$convex of order$m$
The function $x^{m+1}$is convex of order$m$and then formulas (14), (15) show that$R_{n}[f]$is indeed a degree of accuracy$m$Theorem 2 is therefore a consequence of Theorem 1.
6. The preceding considerations also allow us to calculate, in various forms, the factor$R_{n}\left[x^{m+1}\right]$which appears in formula (9). Given formula (5), notation (6) and hypothesis H.1, we have ($m>1$)