Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators

TL;DR

This paper introduces a broad family of Szász-Mirakjan-Durrmeyer operators with new properties, providing complete asymptotic expansions and a localization result for functions of exponential growth.

Contribution

It generalizes existing operators, establishes their asymptotic behavior, and offers a unifying framework for related operators.

Findings

Complete asymptotic expansions for the operators.

Operators preserve constants and monomials of degree j.

Localization results for functions of exponential growth.

Abstract

In this paper we introduce a general family of Sz\'asz--Mirakjan--Durrmeyer type operators depending on an integer parameter $j \in \mathbb{Z}$. They can be viewed as a generalization of the Sz\'asz--Mirakjan--Durrmeyer operators [9], Phillips operators [11] and corresponding Kantorovich modifications of higher order. For $j\in {\mathbb{N}}$, these operators possess the exceptional property to preserve constants and the monomial $x^{j}$. It turns out, that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result which is interesting in itself.