Convergence of Sz\'asz-Mirakyan-Durrmeyer operators having Laguerre-type weight

TL;DR

This paper introduces a new family of Szász-Mirakyan-Durrmeyer operators with Laguerre-type kernels on [0,∞), analyzing their properties, moments, convergence, and eigenfunctions to extend classical approximation results to unbounded domains.

Contribution

The paper presents a novel class of operators with explicit moments, recurrence relations, and convergence analysis, expanding approximation theory on unbounded domains.

Findings

Derived explicit moments and recurrence relations.

Established local and global $L_p$-convergence results.

Analyzed asymptotic behavior and eigenfunctions.

Abstract

In this paper, we introduce a new family of Szasz-Mirakyan-Durrmeyer operators defined on the half-line [0,\infty), constructed using Laguerre-type kernels. We discuss in detail the algebraic structure and analytical properties of these operators. thoroughly investigated. Explicit closed-form expressions for the moments are derived, along with a differential recurrence relation connecting successive moments. Quantitative estimates on compact intervals are obtained, and Weighted approximation results are provided for unbounded functions. Furthermore, the asymptotic behavior of the central moment is analyzed. We establish both local and global $L_p$-convergence results and identify the eigenfunctions associated with these operators. These findings demonstrate the effectiveness of the proposed generalized operators in extending classical approximation results to the unbounded domain.