Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

TL;DR

This paper introduces a new semi-discrete modulus of smoothness that generalizes previous definitions and provides sharper one- and two-sided error estimates for various approximation operators.

Contribution

The paper develops a novel semi-discrete modulus of smoothness, establishing its properties, equivalence with K-functional, and improved error estimates for classical and sampling operators.

Findings

Sharper error estimates than classical moduli of smoothness.

Establishment of a Rathore-type theorem and K-functional equivalence.

Applicability to Bernstein polynomials, Shannon sampling, and generalized sampling operators.

Abstract

In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and two- sided error estimates under non-restrictive assumptions for pointwise linear operators. The proposed results have been proved exploiting the regularization and approximation properties of certain Steklov integrals introduced by Sendov and Popov in 1983. By the definition of semi-discrete moduli of smoothness here proposed, we derive sharper estimates than those that can be achieved by the classical averaged moduli of smoothness ($\tau$-moduli). Furthermore, a Rathore-type theorem is established, and a new notion of K-functional is also introduced showing its equivalence with the semi-discrete modulus of smoothness and its realization. One-sided estimates of approximation can be established for classical operators on bounded domains, such as the Bernstein polynomials. In the case of approximation operators on the whole real line, one-sided estimates can be achieved, e.g., for the Shannon sampling (cardinal) series, as well as for the so-called generalized sampling operators.