Let us consider a function$f=f(x)$real, of a real variable and$\varphi$an approximation function of$f$,

The function$\varphi$is chosen from a certain determined set of approximation functions. For example, one can choose$\varphi$in a set satisfying certain interpolation conditions. We can generally assume that the functions$f,\varphi$are defined on the same set of the real axis, but this assumption is not essential. However, if we want to use the approximate equality (1) to approximate the values ​​of the function$f$by the corresponding values ​​of the function$\varphi$, we must assume that$\varphi$be defined on a set that contains the domain of definition of$\varphi$.

In approximation problems, we generally seek an approximation function.$\varphi$so that the error$f-\varphi$checks certain restrictions imposed by the very nature of the problem under consideration.

But it is important to try to preserve by…

The interpolating function$P\left(x_{0},x_{1},\ldots,x_{n};f\mid x\right)$preserves the sign of the function$f$This property can be easily demonstrated. It also follows from the formula.

where the functions$\varphi_{\alpha},\alpha=0,1,\ldots,n$defined on$[a,b]$are special polygonal functions, having the nodes$x_{\alpha}$,$\alpha=0,1,\ldots,n$independent of the function$f$and given by the formulas

if$\alpha=1,2,\ldots,n-1$. If$\alpha=0,n$, We have

The property of preserving the sign can be stated as follows: from the right-hand side of formula (2), we deduce:

If the function$f$is non-negative, positive, negative, respectively non-positive (on$[a,b]$), the polygonal function (2) is also non-negative, positive, negative respectively non-positive (on$[a,b]$).

If we apply Abel's transformation formula, approximation (1) preserves certain properties of the function's shape$f$.

Obviously, we must first clarify what we mean by the behavior, or more precisely, by a specific behavior, of a function, and then what we mean by preserving that behavior. We will not attempt to define the behavior of a function. We will only consider a few properties that we agree characterize certain behaviors, such as: non-negativity, monotonicity, convexity of a given order, etc. We will always explain what is meant by the fact that the approximation function preserves a given behavior of the function.$f$.

$\S 2$Let 's
start with an example. Consider$f$a function defined on the finite and closed interval$[a,b]$and either

the polygonal function "inserted" along the vertices, or rather the nodes

This function is characterized by its interpolation properties.

and by the fact that it is continuous on$[a,b]$, being linear on each of the partial intervals

determined by the nodes.

which are the divided differences of order 2 on the nodes

The functions$\chi_{\alpha}$are non-concave (usual) and we deduce the following property of the conservation of usual (first-order) convexity (more precisely, non-concavity):

If the function$f$is non-concave, respectively non-convex of order 1, the polygonal function (4) is also non-concave, respectively non-convex of order 1.

While the conservation of sign and monotonicity holds in the strict sense, the conservation of first-order convexity holds only in the broad sense. This means that if the non-negative (or non-decreasing) function$f$is, in particular, positive (or increasing), the polygonal function (2) is not only non-negative (or non-decreasing), but is also positive (or increasing). At the same time, if the non-concave function of order 1$f$is convex of order 1, the polygonal function (2) is only non-concave of order 1 without necessarily being convex of order 1.

We will make one more remark about polygonal functions. These are continuous functions that reduce to a polynomial of degree 1 on each of the intervals

determined by the nodes. They can also be defined as linear combinations of a polynomial of degree 1 and a finite number of functions of the form$|x-\lambda|+x-\lambda$, Or$\lambda\in[a,b]$.

In general, a linear combination of a polynomial of degree$n$and a finite number of functions of the form$(|x-\lambda|+x-\lambda)^{n}$, Or$\lambda\in[a,b]$, is said to be an elementary function of order$n$($n$( = natural number).

Such a function is

Or

For$\alpha=1,2,\ldots,n$.

The functions$\psi_{\alpha}$are non-decreasing (and not identically zero) and we deduce the following property of the preservation of the monotonicity of the function$f$:

If the function$f$is non-decreasing, increasing, decreasing, respectively non-increasing ($x\in[a,b]$), the polygonal function (3) is also non-decreasing, increasing, decreasing, respectively non-increasing (on$[a,b]$).

If we apply Abel's transformation formula once again to the right-hand side of formula (3), we deduce

Or$\chi_{\alpha}$are still polygonal functions having the nodes

and are independent of the function$f$Divided differences are defined by

The properties of these functions are well known. The case$n=-1$corresponds to the preservation of the sign of the function.

There are very important operators of this form, usually called **positive operators**. Such operators are provided, for example, by Fejér sums of Fourier series, by various Fejér interpolation polynomials, etc. Positive operators play a very important role in the theory of polynomial, trigonometric, and other similar approximations.

For the past few years, we have also begun to study the case$n>-1$, for several determined operators. This research began with the observation that the well-known polynomial of SN Bernstein.

This polynomial preserves, on the interval$[0,1]$, the non-concavity of order$n$of the function$f$for everything$n\geq-1$In this case, we can take$E = I = [0,1]$.

For$n\leq m-1$, this conservation is even in the strict sense: therefore, if$f$is convex of order$n\ (\leq m-1)$, the polynomial ( LABEL:5 ) is also convex (and not just non-concave) of order$n$.

It is also worth noting that here it is sufficient to even assume that the function$f$either non-concave of order$n$on the points$\tfrac{\alpha}{m}$,$\alpha=0,1,\ldots,m$. For$n\geq 2$This condition is broader than non-concavity of order$n$on any set containing these points [ 6 ] .

The noted conservation property of the polynomial ( LABEL:5 ) results from the properties of the derivatives of higher-order convex functions and the formula

This function is continuous and has a continuous derivative of order$n-1$(if$n>1$) on$[a,b]$ [ 7 ] .

Previously, we demonstrated the importance of these functions in the theory of approximation of higher-order convex functions [ 5 ] . Today, these functions are also called *splines*. Recently, several authors have studied them and shown their importance in the theory of function approximation. We particularly note the research of IJ

The example of polygonal interpolation functions shows us the way to generalizing the problems of preserving certain well-defined shapes.

Let us consider the operator$F[f\mid x]$defined on the function space$f$, real and of a real variable$x$, defined on a set$E$of the real axis, having its values ​​in the set of functions defined on the set$I$of the real axis. In what follows, we will assume that the operator$F[f\mid x]$is linear. We have the

Definition. We will say that the operator$F[f\mid x]$preserves (on$I$) the non-concavity of order$n$(of the function$f$) if the function$F[f\mid x]$of$x$is non-concave of order$n$(on$I$) for any function$f$non-concave of order$n$(on$E$).

Let us recall that a function$f$is said to be convex, non-concave, polynomial, non-convex, respectively concave of order$n\ (\geq-1)$on$E$, if all the differences divided by order$n+1$,

of this function, on$n+2$any distinct points (or nodes)$x_{1},x_{2},\ldots,x_{n+2}$of$E$, are positive, non-negative, zero, non-positive, respectively negative.

Operator (9) is defined even if$f$is only defined on the points (10).

By supplementing the research of L. Fejér [ 2 ] , we examined the preservation of monotonicity by the polynomial (9) of L. Fejér [ 10 ] . This is the case$n=0$of the conservation of the convexity shape of order$n$.

Among the results obtained, we note the following: if the points (10) are normally distributed in an interval, or if the points (10) are the zeros of the Jacobi polynomial of degree$n$and whose parameters satisfy the inequalities

then the interpolation polynomial (9) preserves the monotonicity of the function$f$in a neighborhood of each of the roots of the derivative of the polynomial

The notion of normally distributed points is due to L. Fejér and is involved precisely in the study of the non-negativity of Fejér's operator (9).

Of the form (7) is also the Lagrange interpolation polynomial. This polynomial enjoys in particular the properties of non-preservation of the convexity shape [ 10 ] .

We can even study, from the point of view of preserving convexity, operators constructed on multiple nodes. Such an operator is, for example, the Lagrange-Hermite polynomial.

We have published relatively few works in our country on the preservation of the shape of convexity of various orders by operators$F[f\mid x]$of the form specified in the given definition. D. Ripeanu began the study and obtained some remarkable results on operators of the form

which preserve the convexity of the function$f$up to a certain given order.

A. Lupaş [ 3 ] studied the conservation of the convexity shape in other contexts.

We have examined [ 9 ] the preservation of convexity by the more general interpolation operator

Or$x_{0},x_{1},\ldots,x_{m}$are$m+1$distinct given points of$E$And$P_{0},P_{1},\ldots,P_{m}$functions defined on$I$.

We can assume that

We have found the necessary and sufficient conditions for the operator ( LABEL:7 ) to retain on$I$the non-concave order$n-1$, the function$f$being non-concave of order$n$on the points ( LABEL:8 ). The numbering ( LABEL:8 ) of the nodes allows these conditions to be written in explicit form.

We have also shown that, if the points ( LABEL:8 ) are given arbitrarily, there exist interpolation polynomials of degree$m$of the form ( LABEL:7 ), which preserve all convexities of order$m-1$on a finite interval$I$of non-zero length [ 9 ] .

The problem of constructing such polynomials has also been studied by D. Ripeanu [ 11 ] .

Consider the polynomial of L. Fejér [ 2 ] , whose second member is the first term of the Lagrange-Hermite polynomial

of the function$f$on double knots

It is the polynomial of least degree which, as well as its derivative, takes the same values ​​as the function$f$, respectively its derivative$f^{\prime}$, on the points ( LABEL:10 ).

But it is clear that we will have to study, for example, the conservation, by trigonometric operators, of convexity with respect to the (linear) set of trigonometric polynomials of a given order.

Another generalization of higher-order convex functions consists of functions that are of order$n$by segments. Such a function has the property that its set can be decomposed into a finite number of consecutive subsets such that, on each, the function is non-concave or non-convex of the same order$n$.

We say that the whole$E$of the real axis is decomposed (into a finite number) into consecutive subsets if

and if, for all$\alpha=1,2,\ldots,m-1$we have

This generalization of higher-order convex functions leads to a definite shape if, for example, we fix the maximum number of consecutive decomposition subsets of the preceding form. We have shown [ 8 ] that Bernstein's polynomial ( LABEL:5 ) also preserves such shapes.

To give an example, we can consider that a function has such a shape if it is non-decreasing on the interval$[a,b)$and non-increasing over the interval$[b,c]$, Or$a<b<c$.

There are also other properties that can be considered as characterizing well-defined patterns of a function. For example, the fact that the divided difference of order$n$of a function, or the total variation of order$n$ [ 5 ] of this function, remain between two given numbers. Such a pattern is still preserved by the polynomials of S. N. Bernstein.

We can further generalize conservation problems by studying operators that transform functions of a given form into functions of another given form. It is always necessary to properly specify the convexity conditions of certain operators generalizing the Bernstein polynomial.

In this section, we intend to briefly outline some problems of preserving appearance that remain to be studied. Research should be directed in several directions, including the following:

Linear interpolation sets constitute an important special case. In this case, it will be necessary to first study the linear operators, in particular those of the form ( LABEL:7 ), which preserve the respective convexity.

Otherwise, we would have problems that are too general, encompassing all sorts of questions that fall outside this framework. Such are, for example, the very interesting problems, the study of which was begun by L. Fejér, concerning the convexity (of various orders) of the partial sums of trigonometric series.

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