1. 1.

   In approximation problems, we usually look for an approximation function.$\varphi$of the given function$f$so that the error$f-\varphi$of the approximate equality

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checks certain restrictions (delimitations, etc.) imposed by the very nature of the problem under consideration.

But it is important to try to preserve, through approximation (1), certain properties of the function's shape$f$In general, the approximation function$\varphi$is chosen from a certain determined set of functions, a set which depends in some way on the function$f$For example, one can choose$\varphi$in a set satisfying certain interpolation conditions.

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 this behavior. We will not attempt to define the behavior of a function. We will only study some properties that we agree characterize certain behaviors, such as: non-negativity, monotonicity, convexity of a given order, etc.
2. To clarify, 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 real functions defined on the set$I$of the real axis. In what follows, unless otherwise stated, we will assume that the operator$F[f\mid x]$is linear and we introduce the

Definition. We will say that the operator$F[f\mid x]$preserves (on$I$) convexity, non-concavity, polynomiality, non-convexity respectively concavity of order$n$(of the function$f$) if the function$F[f\mid x]$of$x$is convex, non-concave, polynomial, non-convex respectively concave of order$n$(on$I$) for any convex, non-concave, polynomial, non-convex, respectively concave function 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$n(\geqq-1)$on$E$if all the differences divided by order$n+1,\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$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, and negative, respectively. The properties of these functions, as well as those of divided differences, are well known. We move from convex, non-concave, and polynomial functions of order$n$to concave, nonconvex, and polynomial functions of order$n$by changing their sign (on$E$) so by going from$f$has$-f$and vice versa. The properties of concavity and non-convexity conservation therefore simply follow (in the case of linear operators)$F[f\mid x]$), properties of preservation of convexity and non-concaveness of the same order. The case$n=-1$This corresponds to the preservation of the sign of the function. Positive operators (and in general, non-negative operators) play a very important role in the theory of polynomial, trigonometric, and other similar approximations. The case$n=0$corresponds to the preservation of monotony and$n=1$to the preservation of the usual convexity.
3. We studied the case$n>-1$for several determined operators. Our research began with the observation that the well-known SN Bernstein polynomial

conserves over the interval$[0,1]$the non-concave order$n$of the function$f$for everything$n\geqq-1$[2]. In this case we can take$E=I=[0,1]$We obtained some general results concerning operators of type (2) [5] and concerning the preservation of sign and monotonicity by the interpolation polynomials of L. Fejér [6], which is also of the same type. We also obtained some analogous results for the Lagrange interpolation polynomial [4], [7].

We will refer to$L\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right)$the Lagrange polynomial (of degree$n$taking the same values ​​as the function$f$on the$n+1$knots$x_{1},x_{2},\ldots,x_{n+1}$.

Because the coefficient of$x^{n}$in the polynomial$L\left(x_{1},x_{2},\ldots,x_{n+1};f\mid x\right)$is equal to$\left[x_{1},x_{2},\ldots,x_{n+1};f\right]$It follows that the property, almost obvious, is that if$x_{1},x_{2},\ldots,x_{n+1}\in E$, this polynomial preserves the non-concaveness of order$n-1$over any interval$I$This property is, moreover, equivalent to non-concavity of order$n-1$But it is easy to demonstrate that this polynomial preserves non-concavity of order$n-2$on the interval$\left[\frac{x_{1}+x_{2}+\cdots+x_{n}}{n},\frac{x_{2}+x_{3}+\cdots+x_{n+1}}{n}\right]$assuming