A characterization of generalized Lipschitz classes by the rate of convergence of semi-discrete operators

TL;DR

This paper characterizes generalized Lipschitz classes by analyzing the convergence rates of Durrmeyer-type semi-discrete sampling operators in $L^p$ spaces, linking approximation theory with function regularity.

Contribution

It provides a comprehensive characterization of Lipschitz classes via convergence rates of sampling operators, including direct and inverse theorems, with applications to signal prediction.

Findings

Established direct approximation results with quantitative estimates.

Used Hardy-Littlewood maximal inequality to weaken kernel assumptions.

Applied theory to improve convergence rates and signal prediction.

Abstract

In this paper, we establish a comprehensive characterization of the generalized Lipschitz classes through the study of the rate of convergence of a family of semi-discrete sampling operators, of Durrmeyer type, in $L^p$-setting. To achieve this goal, we provide direct approximation results, which lead to quantitative estimates based on suitable $K$-functionals in Sobolev spaces and, consequently, on higher-order moduli of smoothness. Additionally, we introduce a further approach employing the celebrated Hardy-Littlewood maximal inequality to weaken the assumptions required on the kernel functions. These direct theorems are essential for obtaining qualitative approximation results in suitable Lipschitz and generalized Lipschitz classes, as they also provide conditions for studying the rate of convergence when functions belonging to Sobolev spaces are considered. The converse implication is, in general, delicate, and actually consists in addressing an inverse approximation problem allowing to deduce regularity properties of a function from a given rate of convergence. Thus, through both direct and inverse results, we establish the desired characterization of the considered Lipschitz classes based on the $L^p$-convergence rate of Durrmeyer sampling operators. Finally, we provide remarkable applications of the theory, based on suitable combinations of kernels that satisfy the crucial Strang-Fix type condition used here allowing to both enhance the rate of convergence and to predict the signals.