On the preservation of functions with exponential growth by modified Ismail–May operators

Abstract

In this study, our aim is to provide a modification of the so‐called Ismail–May operators that preserve exponential functions eAx,.7emA.5em∈.5emℝ. In support of this, we begin by estimating the convergence rate of the operators in terms of usual and exponential modulus of continuity. We also provide a global approximation and a quantitative Voronovskaya result. Moreover, to validate the modification, we exhibit some graphical representations using Mathematica software to compare the original operator and its modification. We conclude that the modified operators not only preserve exponential functions but also provide faster rate of convergence for A > 0.

Convergence estimates of certain gamma type operators

“…In the past few decades, integral operators have become an extensive area of research, and studies have been dedicated towards analyzing their approximation behaviors. We suggest previous articles 11–16 …”

“…We suggest previous articles. [11][12][13][14][15][16] King 17 proposed a modification of Bernstein operators which preserve the test functions e 0 (t) = 1 and e 2 (t) = t 2 . Inspired by this approach, many other mathematicians devoted their research to produce modifications of well-known approximation processes fixing certain (polynomial, exponential, or more general) functions and to study their approximation and shape preserving properties.…”