Special measures of smoothness for approximation by sampling operators in $L_p(\Bbb{R}^d)$

TL;DR

This paper introduces a new measure of smoothness tailored for better $L_p$-error estimates in sampling approximation, addressing limitations of traditional measures especially for less smooth functions.

Contribution

It develops a modified smoothness measure incorporating local behavior at sampling points, enabling precise error estimates and convergence criteria for various sampling operators in $L_p$ spaces.

Findings

Matching direct and inverse error estimates achieved.

Convergence criteria for sampling operators established.

Application to classical and modern sampling operators demonstrated.

Abstract

Traditional measures of smoothness often fail to provide accurate $L_p$-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified measure of smoothness that incorporates the local behavior of a function at the sampling points through the use of averaged operators. With this new tool, we obtain matching direct and inverse error estimates for a wide class of sampling operators and functions in $L_p$ spaces. Additionally, we derive a criterion for the convergence of sampling operators in $L_p$, identify conditions that ensure the exact rate of approximation, construct realizations of $K$-functionals based on these operators, and study the smoothness properties of sampling operators. We also demonstrate how our results apply to several well-known operators, including the classical Whittaker-Shannon sampling operator, sampling operators generated by $B$-splines, and those based on the Gaussian.