Modular convergence of Steklov sampling operators in Orlicz spaces

TL;DR

This paper establishes modular convergence of Steklov sampling operators in Orlicz spaces, extending classical results to a broader functional setting with applications to various specific function spaces.

Contribution

It proves a modular convergence theorem for Steklov sampling operators in Orlicz spaces using a density approach, including new kernel function conditions.

Findings

Modular convergence in Orlicz spaces established

Results extended to L^p, Zygmund, and exponential spaces

Kernel functions satisfying specific conditions are characterized

Abstract

In this paper, we deal with the family of Steklov sampling operators in the general setting of Orlicz spaces. The main result of the paper is a modular convergence theorem established following a density approach. To do this, a Luxemburg norm convergence for the Steklov sampling series based on continuous functions with compact support, and a modular-type inequality in the case of functions in Orlicz spaces has been preliminary proved. As a particular case of general theory, the results in $L^p$, in the Zygmund (interpolation), and in the exponential spaces are deduced. A crucial aspect in the above results is the choice of both band- and duration- limited kernel functions satisfying the partition of the unit property; to provide such examples an equivalent condition based on the Poisson summation formula and the computation of the Fourier transform of the kernel has been employed.