The authors state a result about the local saturation of sequences of linear operators that preserve the sign of -th derivative of the functions.
Let , . Let be the space of all real-valued and -times continuously differentiable functions defined on and be the -th differential operator with the space of all continuous functions defined on and the identity operator, respectively. Besides, denotes the subspace formed by the functions of which are bounded on . A function is said to be -convex if on and a linear operator is said to be -convex if it maps -convex functions onto -convex functions.
Definition. Let denote the second-order linear differential equation . Consider an interval such that has a unique solution, continuous in , taking any given real values and at any two given and within . Assume that and are continuous and differentiable whenever this is required. A real function defined in is said to be in if for every , , such that , being the solution of taking the values and at and .
Further, let be a closed real interval, and satisfying the following shape preserving property and asymptotic formula: (H1) for all , is -convex, (H2) there exist a sequence of real positive numbers, and a function strictly positive on such that for all , -times differentiable in some neighborhood of a point , .
The main result is:
Theorem. Let and be the operators which satisfy conditions H1 and H2, let and let with . Then for (i) is in if and only if
(ii) is in if and only if
Finally, the theorem is applied to the well known approximation operators of Bernstein, Szász-Mirakyan, Mayer-König and Zeller, and Bleimann, Butzer and Hahn.