On Bernstein–Chlodowsky Type Operators Preserving Exponential Functions

Abstract

In this paper, we introduce, analyze, and obtain some features of a new type of Bernstein–Chlodowsky operators using a different technique that is utilized as the classical Chlodowsky operators. These operators preserve the functions (formula presented) and (formula presented). As a first result, the rate of convergence of the operator using an appropriately weighted modulus of continuity is obtained. Later, Quantitative-Voronovskaya type and Grüss–Voronovskaya type theorems for the new operators are presented. Then, we prove that the first derivative of the Bernstein–Chlodowsky operators applied to a function converges to the function itself. Finally, the variation detracting property of the operators is presented. It is proved that the variation seminorm property is preserved. Also, it is shown that the operators converge to (formula presented) in variation seminorm is valid if and only if the function is absolutely continuous. © 2020, Springer Nature Singapore Pte Ltd.