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Local saturation of conservative operators. (English) Zbl 1058.41017

The authors state a result about the local saturation of sequences of linear operators that preserve the sign of k-th derivative of the functions.

Let I, i 0 ={0}. Let C i (I) be the space of all real-valued and i-times continuously differentiable functions defined on I and D i be the i-th differential operator with C(I)=C 0 (I) the space of all continuous functions defined on I and D 0 the identity operator, respectively. Besides, C B i (I) denotes the subspace formed by the functions of C i (I) which are bounded on I. A function fC i (I) is said to be i-convex if D i f0 on I and a linear operator is said to be i-convex if it maps i-convex functions onto i-convex functions.

Definition. Let L(y)=0 denote the second-order linear differential equation L(y)D 2 y+p 1 (t)D 1 y+p 2 (t)y=0. Consider an interval (a,b) such that L(y)=0 has a unique solution, continuous in (a,b), taking any given real values y 1 and y 2 at any two given t 1 and t 2 within (a,b). Assume that p 1 (t) and p 2 (t) are continuous and differentiable whenever this is required. A real function f(t) defined in (a,b) is said to be sub-(L) in (a,b) if f(t)S(f,t 1 ,t 2 )(t) for every t, t 1 , t 2 such that a<t 1 <t<t 2 <b, S(f,t 1 ,t 2 ) being the solution of L(y)=0 taking the values f(t 1 ) and f(t 2 ) at t 1 and t 2 .

Further, let I be a closed real interval, k 0 and L n :C k (I)C k (I) satisfying the following shape preserving property and asymptotic formula: (H1) for all n, L n is k-convex, (H2) there exist a sequence λ n of real positive numbers, and a function pC k (I) strictly positive on Int(I) such that for all gC B k (I), k+2-times differentiable in some neighborhood of a point xInt(I), lim n λ n (D k L n g(x)-D k g(x))=D k (pD 2 g)(x).

The main result is:

Theorem. Let L n and L be the operators which satisfy conditions H1 and H2, let M0 and let a,bInt(I) with a<b. Then for f,wC B k (I) (i) MD k w+D k f is sub-(L) in (a,b) if and only if

D k L n f(x)-D k f(x)-M(D k L n w(x)-D k w(x))+σ(λ n -1 ),x(a,b)·

(ii) MD k w-D k f is sub-(L) in (a,b) if and only if

D k L n f(x)-D k f(x)M(D k L n w(x)-D k w(x))+σ(λ n -1 ),x(a,b)·

Finally, the theorem is applied to the well known approximation operators of Bernstein, Szász-Mirakyan, Mayer-König and Zeller, and Bleimann, Butzer and Hahn.


MSC:
41A40Saturation (approximations and expansions)