On the convergence properties of Durrmeyer-Sampling Type Operators in Orlicz spaces

TL;DR

This paper studies the convergence behavior of Durrmeyer sampling operators in Orlicz spaces, providing theoretical results on pointwise, uniform, and modular convergence, along with applications and graphical illustrations.

Contribution

It offers a unified framework for analyzing convergence of Durrmeyer sampling operators in various function spaces, including Orlicz and specific Lebesgue spaces, with quantitative approximation estimates.

Findings

Established pointwise and uniform convergence theorems.

Proved modular convergence in Orlicz spaces.

Provided applications with graphical examples.

Abstract

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces $L^\varphi(\mathbb{R})$. From the latter result, the convergence in $L^p(\mathbb{R})$-space, $L^\alpha\log^\beta L$, and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.