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

TL;DR

This work addresses characterizing generalized Lipschitz classes via the $L^p$-convergence rate of semi-discrete Durrmeyer sampling operators $\mathcal{D}_w^{\varphi,\psi}$. It develops direct approximation results through $K$-functionals and moduli of smoothness, and extends to settings where the Hardy-Littlewood maximal inequality relaxes kernel assumptions, enabling robust $L^p$-convergence estimates. An inverse theorem shows that a rate $\|\mathcal{D}_w^{\varphi,\psi}f-f\|_p=\mathcal{O}(w^{-\alpha})$ with $0<\alpha<r$ implies $f\in Lip^*(\alpha,L^p)$, yielding a full equivalence between convergence rate and regularity. The paper also provides practical kernel constructions (e.g., B-splines, Jackson kernels) that satisfy the Strang-Fix conditions, demonstrating enhanced convergence and prediction capabilities for generalized sampling and Kantorovich-type variants.

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.

Figures (1)

• Figure 1: The graphs of the kernels $\sigma_3$ (dashed blue line) and $\tau$ (solid red line).

Theorems & Definitions (20)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2: DurrConv
• Remark 3.1
• Theorem 3.1
• proof
• Corollary 3.1
• Theorem 3.2
• proof
• Theorem 3.3