1. 1.

   Let us consider the linear operator

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defined on the function space$f=f^{\prime}(x)$, real and of a real variable$x$, defined on a linear set$E$containing the points$x_{0},x_{1},\ldots,x_{m}$We can assume that the points$x_{i}$are distinct and are numbered in ascending order of magnitude, therefore

The functions
(3)

characterize the operator (1) and are real, of the real variable$x$defined on a non-zero interval$I$.*) For all$f$the value of operator (1) is a function defined on the interval$I^{**}$.

Expression (1) represents and can therefore be called an interpolation function of the function$f$The points (2) are the corresponding interpolation nodes.

If the functions (3) are polynomials, the operator (1) is a (generalized) interpolation polynomial of the function$f$. In this case the degree of the interpolation polynomial is equal to the largest of the degrees of the polynomials (3).

The designation of function, or interpolation polynomial respectively, is justified by the fact that on any common point$x_{0}$sets$E$And$I$, the value$F\left[f\mid x_{0}\right]$of function (1) can be considered as an approximate value of$f(x_{0})$This approximation is obtained by the interpolation process expressed by the operator (1).

Here we understand interpolation in a generalized sense. The function (1) is a proper interpolation function (an interpolation polynomial) if the interval$I$contains the nodes (2) and if the functions$F[f\mid x],f$coincide on the nodes, regardless of the function$f$This last condition is expressed by the conditions

Of this form are, for example, the Lagrange interpolation polynomial, various types of interpolation polynomials due to I. FEJÉR [1], etc. But there are also interpolation polynomials that do not satisfy conditions (4) and yet are of great importance. Such is the well-known polynomial of SN Bernstein

In this case we have

1. 2.

   The polynomial (5) has the important property that it preserves the sign of the function$f$on the interval$[0,1]$Therefore, for any function$f$non-negative (on$E$), it is non-negative on$[0,1]$.

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I have previously demonstrated [2] that polynomials (5) also possess other properties of preserving the shape of the function$f$If the function$f$is non-decreasing or, in general, if it is non-concave of order$n$(on$E$) the same is true for the polynomial (5) on [0,1]. The polynomials (5) of SN Bernstein therefore preserve (on [0,1]) the non-concavity of any order ($\geqq-1$) of the function$f$.

We can introduce 1a

Definition 1. We will say that the interpolation function (1) preserves (on the interval$I$) the non-concavity of order$n$(of the function$f$) if, the function (1) is non-concave of order$n$(on I) for any function$f$nonconcave of order$n$(on$E$3.
Recall that a function is said to be non-concave of order$n(\geqq-1)$on$E$if all the differences divided by order$n+1$of this function, on$n+2$any (distinct) points of$E$, are non-negative. If all these divided differences are positive, the function is said, in particular, to be convex of order$n$on$E$Non-concavity and convexity of order -1 are equivalent to non-negativity and positivity of the function, respectively. Non-concavity and convexity of order 0 are equivalent to non-decreasing and increasing of the function, respectively. Finally, non-concavity and convexity of order 1 are equivalent to the function's usual non-concavity and convexity, respectively.

If the differences divided by order$n+1$of the function are all non-positive respectively all negative, we say that this function is non-convex respectively concave of order$n$(on$E$). There are specifications analogous to those above in the cases$n=-1$, 0 or 1.

If the function$f$is non-concave respectively convex of order$n$, the function -$f$is non-convex respectively concave of order$n$and vice versa.

A function whose differences are all divided by order$n+1$zero is said to be a polynomial of order$n$(on$E$). For a function to be polynomial of order$n$It is necessary and sufficient that it be both non-concave and non-convex of order$n$.

Convexity (concavity) and polynomiality are special cases of non-concavity (non-convexity) of the same order. If a function is non-concave of order$n\geqq 0$on the non-zero interval$I$but if it is not convex of order$n$(on$I$), there exists a non-zero subinterval of$I$on which the function is polynomial of order$n$This property is obviously not true for$n=-1$.

If the whole$E$is formed by at most$n+1$points ($n\geqq 0$), any function defined on$E$is polynomial of order$n$(on$E$).

A polynomial of degree$n$is a function of the form

THE$a_{i}$(the coefficients of the polynomial) being arbitrary constants. If$a_{0}\neq 0$The first polynomial (6) has effective degree$n$Any polynomial function of order$n(\geqq 0)$is a polynomial of degree$n^{*}$The (identically) zero function is a polynomial function of any order$n\geq-1$, therefore a polynomial
*) To simplify, we can, without any inconvenience, consider a polynomial to be a polynomial function, although the two notions are logically distinct.
A polynomial of any degree$n\geqq-1$We can assume that the effective degree of this polynomial is equal to -1.

We refer to$\left[\xi_{1},\xi_{2},\ldots,\xi_{n+1};f\right]$the difference divided by order$n$and by$L\left(\xi_{1},\xi_{2},\ldots,\xi_{n+1};f\mid x\right)$the Lagrange polynomial, therefore the polynomial taking the values ​​of the function$f$on the nodes$\xi_{i},i=1,2,\ldots$,$n+1$.

In what follows, we will use several properties of higher-order convex functions, divided differences, and Lagrange polynomials. These properties are generally known. The most important ones will be briefly recalled as they appear.
4. We aim to find conditions that the functions (3) must satisfy for the interpolation function (1) to preserve non-concavity of order$n$of the function$f$.

Before going further, let us note that we can give a definition analogous to definition 1 for the preservation of non-convexity and polynomiality. Moreover, any interpolation function (1) that preserves non-concavity also preserves non-convexity of the same order, and vice versa.

In particular, therefore,
THEORIM 1. If the interpolation function (1) preserves non-concavity of order n, it reduces to a polynomial of degree n for any polynomial$f$degree$n^{*)}$.

We also have
THEOREM 2. If$n\geqq m$, so that the interpolation function (1) preserves the non-concavity of order$n$, it is necessary and sufficient that the functions (3) reduce to polynomials of degree$n$.

Indeed, the polynomial

is a non-concave function of order$n$, whatever the constant$\lambda$(because of inequality)$m\leqq n$For this function we have$F[f\mid x]=\lambda P_{i}$and it is necessary and sufficient that$P_{i}$And$-P_{i}$be non-concave of order$n$Therefore, that$P$let a polynomial of degree$n$.

It follows that if the interpolation function (1) preserves the non-concavity of any order$n\geqq-1$, the functions (3) reduce to polynomials of degree$m$and, moreover, the function$F[f\mid x]$reduces to a polynomial of degree$n(\geqq-1)$if$f$is a polynomial of degree$n$An interpolation function (1) that preserves non-concavity of any order is therefore a polynomial of degree$m$who never raises his degree so$f$is a polynomial.

5. It is easy to see that there exist interpolation functions (1) preserving non-concavity of any order$n\geqq-1$To obtain such an interpolation function, it suffices to take any non-negative constants for the functions (3). We can even state that any interpolation polynomial of degree$k$preserves the non-concaveness of any order$n\geqq k$.
6. For the interpolation function (1) to preserve the non-concavity of order - 1 (non-negativity) it is necessary and sufficient that the functions (3) be non-negative.

Let us now suppose that$0\leqq n\leqq m-1$. According to a formula for the transformation of divided differences [5] we have
(7)$F[f\mid x]=\sum_{i=0}^{n}\left[x_{0},x_{1},\ldots,x_{i};f\right]Q_{i}+\sum_{i=0}^{nn-1}\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right]R_{n,i}$oì
(8)$\quad Q_{i}=\sum_{j=i}^{m}\left(x_{j}-x_{0}\right)\left(x_{j}-x_{1}\right)\ldots\left(x_{j}-x_{i-1}\right)P_{j},\quad i=0,1,\ldots,m$
(9)$\quad R_{n,i}=\left(x_{i+n+1}-x_{i}\right)\sum_{j=i+n+1}^{n}\left(x_{j}-x_{i+1}\right)\left(x_{j}-x_{i+2}\right)\cdots\left(x_{j}-x_{i+n}\right)P_{j}$,

Here is the product$\left(x_{j}-x_{0}\right)\left(x_{j}-x_{1}\right)\ldots\left(x_{j}-x_{i-1}\right)$if$i=0$and the product$\left(x_{j}-x_{i+1}\right)\left(x_{j}-x_{i+2}\right)\ldots\left(x_{j}-x_{i+n}\right)$if$n=0$are replaced by 1.

We then have
THEOREM 3. If$0\leqq n\leqq m-1$, so that the interpolation function (1) preserves on I the non-concavity of order$n$of any function$f$non-concave of order$n$Regarding points (2), it is necessary and sufficient that the functions$Q_{0}, Q_{1}, ..., Q_{n}$let polynomials of degree$n$And$R_{n,0},R_{n,1},\ldots,R_{n,mn-1}$nonconcave functions of order$n$(on I).

The property is easily derived by noting that one can find a non-concave function of order$n$on points (2) such that$\left[x_{0},x_{1},\ldots,x_{i};f\right],i=0,1,\ldots,n$take on any real values ​​and$\left[x_{i},x_{i+1},\ldots,x_{i+n+1};f\right],i=0,1,\ldots,mn-1$any non-negative real values. It must also be taken into account that any linear combination of non-concave functions of order$n$, with non-negative coefficients, is also a non-concave function of order$n$.

If the functions (3) are ($n+1$)-times differentiable, therefore, in particular, if they are polynomials, the conditions of Theorem 3 can be written

Indeed, for a function ($n+1$)-times differentiable or non-concave of order$n$, it is necessary and sufficient that its derivative of order$n+1$either non-negative. Finally, it is easy to see that from the above also follows the

THEOREM 4. For the interpolation function (1) to preserve (on I) any non-concavity of order$n\geqq-1$functions$f$non-concave of order$n$Regarding points (2), it is necessary and sufficient that$Q_{i}$let be a polynomial of degree i for$i=0,1,\ldots,m$and that for all$n\geqq-1$And$i=0,1,\ldots,mn-1$the polynomial$R_{n,i}$either non-concave of order$n$(on I).

Note that because$Q_{i}$is a polynomial of degree$i$For$i=0,1,\ldots,m$It follows that the functions (3) are polynomials of degree$m$To complete the formulas (9) we take$R_{-1,i}=P_{i},i=0,1,\ldots,m$and we see that$R_{n,i}$are all polynomials of degree$m$The conditions of the statement can also be written

(13)$\quad R_{n,i}^{(n+1)}\geqq 0,i=0,1,\ldots,mn-1,n=-1,0,1,\ldots,m-1$7.
In Theorems 3 and 4, it is essentially assumed that the domain of definition$E$of the function$f$reduces to the set of points (2). The non-extendability of non-concave functions of order$>1$[3] shows us that if$E$is any set containing the points (2) we can only affirm that the conditions of the theorems are merely sufficient. We therefore have 1es

THEOREM 3'. If$0\leqq n\leqq m-1$, so that the interpolation function (1) preserves the non-concavity of order$n$it is enough that$Q_{0}, Q_{1}, ..., Q_{n}$let polynomials of degree$n$and that the functions$R_{n,0},R_{n,1},\ldots,R_{n,nn-1}$be non-concave of order$n$Theorem 4'. For
the interpolation function (1) to preserve all non-concavities of order$n\geqq-1$it is enough that$Q_{i}$let a polynomial of degree$i$For$i=0,1,\ldots,m$and that for$i=0,1,\ldots,mn-1,n==-1,0,1,\ldots,m-1$the polynomial$R_{n,i}$either non-concave of order$n$8.
We know that the non-concave functions of order$-1.0$or 1 are always and everywhere extendable. We therefore deduce the

THEOREM 5.$1^{\circ}$. For the interpolation function (1) to retain non-negativity it is necessary and sufficient that the functions (3) be non-negative.
$2^{\circ}$For the interpolation function (1) to maintain non-decreasing, it is necessary and sufficient that the sum$\bar{P}_{0}+P_{1}+\ldots+P_{m}$reduces to a constant and the functions

are non-decreasing.
$3^{\circ}$For the interpolation function (1) to retain the usual non-concavity (of order 1), it is necessary and sufficient that the sums

reduce to polynomials of degree 1 and that the functions

are non-concave of order 1.
We have taken into account formulas (8) and (9) and the numbering (2) of the nodes.

It is easy to find similar statements for$m=0$And$m=1$.
When$n>1$The necessary and sufficient conditions are more complicated. When$n$is random but$m=n+1$the conditions of the theorem$3^{\prime}$are still necessary (even if$n>1$Indeed, in this case any non-concave function of order$n$on the nodes is always and everywhere extendable, notably by the Lagrange polynomial$L\left(x_{0},x_{1},\ldots,x_{n+1};f\mid x\right)$9.
By introducing a slight restriction through the modification of definition 1, more interesting results can be obtained. This modification consists of requiring the preservation not only of non-concavity, but also of convexity.

We introduce 1a
Definition 2. We will say that the interpolation function (1) preserves (on the interval$I$) the convexity of order$n$(of the function$f$) if the function (1) is convex of order$n$(on I) for any function$f$convex of order$n$(on$E$).

If we take into account that the limit of a convergent sequence of non-concave functions of order$n$is also a non-concave function

Conversely, we have

Or$\varphi_{i}=\left(x-x_{0}\right)\left(x-x_{1}\right)\cdots\left(x-x_{i-1}\right),i=0,1,\ldots,m\left(\varphi_{0}=1\right)$,

We also have$R_{-1,i}=\bar{R}_{-1,i}=P_{i},i=0,1,\ldots,n$

1. 12.

   We will demonstrate that if the nodes (2) are given arbitrarily, there exist interpolation polynomials (1) of degree$m$which retain all convexities of order$\leqq m-1$, therefore which also retain all the non-concavities of order$\geqq-1$, on a finite, non-zero interval$I$We will refer to it as$a,b,a<\bar{b}$the extremities of$I$.

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We will construct an interpolation polynomial of the desired form using the

Lemma 1. If I is a finite interval, with endpoints$a,b,a<b$, if$0\leqq i\leqq m$and if$S_{0},S_{1},\ldots,S_{i+1}$are$i+2$polynomials of degree$m$we can find a polynomial$P$degree$m$such as one has

on$I^{*}$).
*) A slightly more general property can be stated in the following form:
If$I$is a finite interval and if$Z_{0},Z_{1},\ldots,Z_{i+1}$are$i+2$($i\geqq 0$) polynomials, the last of which is of degree$k$we can find a polynomial$P$degree$k+i+1$such as one has

on$I$If
the polynomial$Z_{i+1}$is of effective degree$k$, we can choose, from among the infinite number of solutions to the problem, a polynomial$P$of effective degree$k+i+1$.

Indeed, the polynomial