APPROXIMATION OF DERIVATIVES BY NONLINEAR OPERATORS

. There are obtained two theorems on simultaneous approximation, by using generalized convex operators.


INTRODUCTION
In [6] Tiberiu Popoviciu obtained that Bernstein operators preserve convexity of higher orders.On the other hand the sequence of Bernstein operators has the property of the uniform approximation of the derivatives of higher order.This is a fact more general.Sendov and Popov obtained in [8] that, roughly speaking, if a sequence of linear positive operators that preserve convexity of higher orders has the property of the uniform approximation of continuous functions then it has also the propriety of the uniform approximation of derivatives of higher orders on any compact subinterval strictly contained in the interval of definition of functions.
In this paper we shall obtain two theorems concerning the uniform approximation of derivatives of higher orders by using sequences of nonlinear operators having the propriety of preservation of some type of generalized convexity of higher orders.As regard to [8] our scheme of the proof is simplified, but it requires a supplementary order of derivability.
In a similar mode, by replacing in (1) the inequality "≥" by ">", or "=" one can define the functions that are c-convex, respectively c-polynomial of order m.Denote by K m [a, b], F the space of functions that are c-nonconcave of order m.
Remark.In the case F = R a function is c-nonconcave of order m ≥ −1 if and only if it is either usual nonconcave of order m or it is usual nonconvex of order m (see [5]).
By considering a and b fixed one obtains the minimum value of max{ a, v , b, v } in the case λ 1 = −1 and λ 2 = . . .= λ p = 0 and it is equal to There is a number Define g := f n k , y := x k and ) dt, Here it is used the Riemann integral for functions with values in an Euclidean space. Since By approximating I 1 and I 2 by Riemann sums we obtain First consider the case I 1 = 0 and One obtains a contradiction.In the case I 2 = 0 one obtains as above that g(y The case I 1 = 0 is similar.Theorem is proved.
Remark.In the case F = R the result in Theorem 3 is given in [8].
(For j = 1 take j i=2 = 1).Using the relation one can proved by induction with regard to p that We have The main result of this section is the following one.
. This one is a consequence of the Peano's formula: We can choose by induction the numbers λ j > 0, 2 ≤ j ≤ k + 1, such that: (for j = k + 1 take k+1 i=j+1 = 0).Let v ∈ F with v = 1, and consider the function: Indeed, let 2 ≤ j ≤ k + 1 and two sets of distinct points of I: x 0 , . . ., x j and y 0 , . . ., y j .Using the inequalities above one obtains: From Theorem 5 it follows (L n (h)) (j) , and from Theorem 3 it can deduce by induction, for 1 ≤ j ≤ k: From these limits it follows (6).

CONVEX OPERATORS FOR APPROXIMATION OF REAL-VALUED FUNCTIONS
Recall that for n ≥ 1, a subset Z ⊂ C[a, b] is named n-parameter family if for any distinct points x i ∈ [a, b], 1 ≤ i ≤ n, and any real numbers y i , 1 ≤ i ≤ n there is an unique ψ ∈ Z such that ψ(x i ) = y i , 1 ≤ i ≤ n.Convexity with regard to a n-parameter family was introduced by Tiberiu Popoviciu in [7] and was extensively studied by L. Tornheim in [9] and E. Popoviciu in [3] (and in others).Definition 8. [7].
We consider the following definition.The main result of this section is the following.
Theorem 10.Let k ≥ 1 and for each 2 ≤ j ≤ k + 1 let Z j ⊂ C k+1 [a, b] be a j-parameter family.Suppose that there are the numbers M j > 0 such that • .Denote respectively by F [a, b], F the space of functions defined on [a, b] and with values in F , by C [a, b], F the subspace of continuous funtions, endowed with the Chebysev norm • [a,b] and for the integer m ≥ 1 denote by C m [a, b], F the subspace of m times continuously derivable functions.In the the case F = R we omit to write F .If x 0 , x 1 , . . ., x m+1 , m ≥ −1 are distinct points of [a, b] then, for a function f : [a, b] → F denote by [f ; x 0 , x 1 , . . ., x m+1 ] the divided difference of function f on the points x i , 0 ≤ i ≤ m + 1.We introduce the following definition:

Definition 6 .
Afterwards the theorem results by induction.Now consider the following definition.An operator L

Definition 9 . 7 )
Let Z ⊂ C[a, b] be a n-parameter family, n ≥ 1.An operator L : C[a, b] → F[a, b] is Z-convex if the following conditions are verified(If f ∈ C[a, b] is Z-convex then L(f ) is usual convex of order n − 1 If f − g is Z-convex , f, g ∈ C[a, b],(8) then L(f ) − L(g) is usual convex of order n − 1.