Approximation of Real Functions by a Generalization of Ismail–May Operator

Abstract

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth.

Approximation properties of a discrete operator

“…where α > 0, β ≥ 0, k ∈ N ∪ {0}, x ∈ [0, ∞) and f ∈ C[0, ∞)(the space of all continuous functions on [0, ∞)). Our interest in studying more approximation properties and error estimation of the concerned operator was sparked by Holhos' groundbreaking work in his publication [6].…”

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In the present research, we estimate some approximation properties for the generalization of a sequence of positive linear operators considered by Holhoş [6] with the following operators available in literature as special examples: Szàsz operator, Ismail-May operator associated with x(1 + x) 2 . We estimate its recurrence relation, moments and verify the same for its particular cases.

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