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

Danilo Costarelli, Donato Lavella

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 ($τ$-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.

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

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 (-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.

Paper Structure

This paper contains 7 sections, 33 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: moduli of smoothness, $\tau$-moduli and Steklov functions
  3. A new semi-discrete modulus of smoothness: one- and two- sided error estimates
  4. K-functionals: equivalence and realization
  5. Examples and measures of smoothness
  6. One-sided estimates for approximation operators
  7. Conclusions

Key Result

Theorem 2.5

For $f \in M(\mathbb{R}) \cap L^p(\mathbb{R})$, $1 \leq p < \infty$, and $k \in \mathbb{N}$, one has While, for $f \in M([a,b])$ we get:

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: ButzerStens, Proposition 2.7 and SendovPopov, Theorem 1.4
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 51 more