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Modular convergence of Steklov sampling operators in Orlicz spaces

Danilo Costarelli, Erika Russo

TL;DR

The paper extends Steklov sampling operators from classical $L^p$ settings to the broad Orlicz space framework $L^(\mathbb{R})$, proving a modular convergence theorem via a density-based approach grounded in the Luxemburg norm and modular inequalities. It establishes that $S_w^r f$ is well-defined in $L^(\mathbb{R})$, is modularly continuous, and converges modularly to $f$ for all $f\in L^(\mathbb{R})$, with norm convergence for compactly supported continuous functions. The authors also provide concrete corollaries for standard spaces like $L^p$, Zygmund spaces, and exponential spaces, and demonstrate the applicability with representative kernels (Fejér, Jackson-type, and central B-splines). The work offers practical guidance on kernel choice and yields a unified framework for convergence in broad function spaces, setting the stage for quantitative error estimates via modular and norm tools in future work.

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.

Modular convergence of Steklov sampling operators in Orlicz spaces

TL;DR

The paper extends Steklov sampling operators from classical settings to the broad Orlicz space framework , proving a modular convergence theorem via a density-based approach grounded in the Luxemburg norm and modular inequalities. It establishes that is well-defined in , is modularly continuous, and converges modularly to for all , with norm convergence for compactly supported continuous functions. The authors also provide concrete corollaries for standard spaces like , Zygmund spaces, and exponential spaces, and demonstrate the applicability with representative kernels (Fejér, Jackson-type, and central B-splines). The work offers practical guidance on kernel choice and yields a unified framework for convergence in broad function spaces, setting the stage for quantitative error estimates via modular and norm tools in future work.

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 , 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.
Paper Structure (7 sections, 10 theorems, 97 equations)

This paper contains 7 sections, 10 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Orlicz spaces and modular convergence
  3. Steklov sampling operators and preliminary results
  4. Modular convergence in Orlicz spaces
  5. Particular cases and examples
  6. Applications and further examples
  7. Future works and open problems

Key Result

Lemma 3.2

Let $\chi \colon \mathbb{R} \to \mathbb{R}$ be a given function with $\chi \in L^1(\mathbb{R})$, satisfying $(\chi_1)$. In addition, assuming that the function $g(x):= -(ix)^j \chi(x)$, $j \in \mathbb{N}$($x \in \mathbb{R}$ and $i$ denotes the complex unit) and belongs to $L^1(\mathbb{R})$, the foll

Theorems & Definitions (18)

  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • Remark 3.5
  • Theorem 4.1
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 8 more