1. 1.

   This first communication is dedicated to the memory of Dimitrie Pompeiu. Our illustrious predecessor, D. Pompeiu, in addition to his highly original results, knew how to show us how seemingly minor facts can lead to remarkable theories.
   D. Pompeiu placed great emphasis on the famous mean value theorem .

   Report issue for preceding element

($f$continue on$[a,b]$, derivable on$]a,b[$), by suggesting various problems and generalizations, several of which have been solved by Romanian or foreign mathematicians.

In particular, he was interested in the position of the point$\xi$in the interval ]$a,b$[when the function belongs to certain particular sets, such as, for example, the set of polynomials of a given degree.

I do not intend to present a history of the so-called "contraction interval" problems, but only to point out a result I recently obtained which shows that the problems raised by D. Pompeiu can be further generalized.
2. Consider a linear (additive and homogeneous) functional$R(f)$, defined on a linear set$S$of functions$f$real, defined and continuous on an interval (of non-zero length)$I$of the real axis. We will assume that$S$Contains all polynomials.

I remind you that the entire$m\geqslant-1$held by the property that

(For$m=-1$, only$R(1)\neq 0$) is called the degree of accuracy of$\bar{R}(f)$and that the linear functional$R(f)$is said to be of simple form if for all$f\in S$we have.

Or$K$is a number$\neq 0$and independent of the function$f$And$\xi_{i}$are$m+2$distinct points$I$which generally depend on the function$f$Moreover, if$m\geqslant 0$we can always take the points$\xi_{i}$, inside$I$.

The second member of (2) also includes the divided difference of the function$f$on the nodes$\xi_{1},\xi_{2},\ldots,\xi_{m+2}$and whose definition is well known.

In particular, if$m\geqslant 0,R(f)$is the degree of accuracy$m$, of the simple form and if$f\in S$has a derivative$f^{(m+1)}$order$m+1$inside$I_{\text{, }}$we have

Or$\xi\in\operatorname{int}I$Cauchy
's classic formula

Or$\min_{i}(x)\leqq\xi\leqq\max_{i}\left(x_{i}\right)$and which generalizes formula (1) is in turn a special case of formula (3).
3. Regarding formula (2), one can obtain results of the "Pompeii type".

Let us assume that the following assumptions are true.

1. 1.

   mi is a non-negative integer.

   Report issue for preceding element
2. 2.

   S is formed by all the functions$f$having a continuous derivative of order$m+1$on the bounded and closed interval$[a,b]$.

   Report issue for preceding element
3. 3.

   $R(f)$is a linear functional defined on$S$, degree of exactness$m$of a simple and bounded form with respect to the norm

   Report issue for preceding element

for an integer$k$such as$0\leqq k\leqq m+1$4.
Point e is given by

and we have$a<0<b$5.
The function$f$is non-concave of order$m+1$and not concave in order$m+2$.

Then the formula for the average (3) is verified for a$\xi\in[c,b]$In the case
of Cauchy's formula (4) we have$c=\frac{1}{m+2}\sum_{i=1}^{m+2}x_{i}$For the demonstration
and other similar properties see my previous work [2].

Certain restrictions imposed on the derivative$f^{(m+1)}$and the position of the point$\xi$arise from the method of demonstration we have employed and which can certainly be resolved.
4. D. Pompeiu has also demonstrated, by analogy with the classical formula (1), that we have

if$x_{1},x_{2},\xi$and the function$f$satisfy (for$n=1$) under the conditions under which formula (6) takes place.

A former student, Carol Szász, a member of the Analysis Circle at the University of Cluj, extended D. Pompeiu's result by demonstrating the following property.

Either$C$the coefficient of$x^{r}$in the Lagrange-Hermite interpolation polynomial$L\left(x_{1},x_{2},\ldots,x_{n+1};f\right)(n>0,0\leqq r\leqq n)$on the knots$x_{1},x_{2},\ldots\ldots,\hat{w}_{n+1}$assumed not all confused and relative to the function$f$. If$f$is a continuous function on a positive interval b$T(\infty\in I\Rightarrow x>0)$and has a derivative$f^{(in)}$of order n on int$I$Aloys ,
we have

where belongs to the interior of the smallest interval containing the points$x_{1},x_{2},\ldots,x_{n+1}$.

We consider the Lagrange-Hermitè polynomial ordered according to successive powers of$x$and the interval$I$is of non-zero length.
5. The previous property can be generalized as follows.

If the following conditions are met:

1. 1.

   $x_{1},x_{2},\ldots,x_{n+1}$are$n+1(n>0)$points not all combined of interval I.

   Report issue for preceding element
2. 2.

   $f$is a continuous function on$I$and has a derivative$n^{\text{idne }}f^{(n)}$on int$I$.

   Report issue for preceding element
3. 3.

   For simplicity, we denote the interpolation polynomial by L.$L\left(x_{1},x_{2},\ldots,w_{n+1};f\right)$and by

   Report issue for preceding element

The same polynomial for$x_{1}=x_{2}=\ldots=x_{n\uparrow 1}=\xi$
4 . $y_{1},y_{2},\ldots,y_{r+1}(0\leqq r\leqq n)$are$r+1$points on the real axis satisfying the inequalities

So there is a point$\xi$within the smallest interval that contains the points$x_{1},x_{2},\ldots,x_{n+1}$, such as one might have

The points$y_{1},y_{2},\ldots,y_{r+1}$are not necessarily distinct.
For$y_{1}=y_{2}=\ldots=y_{r+1}=0$formula (8) is equivalent to formula (6) and for$r=n$to Cauchy's formula (4).
6. Although formulas (6) and (8) are special cases of other, much more general formulas due to E. Popoviciu [1], they are of interest because they are related to various developments of the Lagrange-Hermite polynomial. A direct elementary proof of formula (8) can be given. We can follow the method that previously allowed us to establish various formulas for the mean of divided differences.

We can assume$x_{1}<x_{2}<\ldots<x_{n+1}$There is then a$\xi\in]x_{1},x_{n+1}\left[\right.$such as any neighborhood of$\xi$contains the points$x_{1}^{\prime}<x_{2}^{\prime}<<\ldots<\omega_{n+1}^{\prime}$such as

The property results from the average formula

Or$x_{1}<x_{2}<\ldots<x_{n+2},1<s<n+2$And$A,B$are positive coefficients, independent of the function$f$And$A+B=1$.

The positivity of the coefficients$A,B$and the formula$A+B=1$result from their explicit values,

Or$\omega(x)=\prod_{i=1}^{n+2}\left(x-w_{i}\right)$.

Because of hypothesis (7), the four divided differences that intervene here are different from 0 and of the same sign (because of the convexity where the concavity of order$r-1$functions that come into play.