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Approximation by linear sampling operators in Banach spaces

Yurii Kolomoitsev

TL;DR

This work develops a comprehensive theory of approximation by linear sampling operators in general Banach lattices $X$, establishing direct and inverse estimates, convergence criteria, and strong converse inequalities that extend beyond classical $L_p$ spaces. By leveraging Steklov averaging and tailored moduli of smoothness, the authors connect sampling-operator errors to semi-discrete $K$-functionals and smoothness functionals, yielding a unified framework for both periodic and non-periodic settings. The results cover a broad class of sampling operators, including Lagrange-type and quasi-interpolation schemes, and provide concrete examples with Bochner–Riesz, Fejér, and sinc-type kernels, together with precise conditions under which the bounds hold. The practical impact lies in enlarging the class of Banach-lattice function spaces and sampling schemes for which sharp, verifiable approximation guarantees can be obtained, with explicit inverse-type and strong-converse statements. Overall, the paper advances the theoretical foundations of sampling-based approximation by unifying direct, inverse, and strong-converse results via $K$-functionals and Steklov-based smoothness across periodic and non-periodic domains.

Abstract

This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special $K$-functionals and their realizations by sampling operators, as well as strong converse inequalities, which, to the best of our knowledge, have not been previously established for sampling operators even in the classical spaces $L_p$. The results extend several classical theorems previously known mainly in $L_p$ and apply to all functions $f\in X$ for which the corresponding sampling operator is well defined, thereby substantially enlarging the class of functions that can be considered in this framework.

Approximation by linear sampling operators in Banach spaces

TL;DR

This work develops a comprehensive theory of approximation by linear sampling operators in general Banach lattices , establishing direct and inverse estimates, convergence criteria, and strong converse inequalities that extend beyond classical spaces. By leveraging Steklov averaging and tailored moduli of smoothness, the authors connect sampling-operator errors to semi-discrete -functionals and smoothness functionals, yielding a unified framework for both periodic and non-periodic settings. The results cover a broad class of sampling operators, including Lagrange-type and quasi-interpolation schemes, and provide concrete examples with Bochner–Riesz, Fejér, and sinc-type kernels, together with precise conditions under which the bounds hold. The practical impact lies in enlarging the class of Banach-lattice function spaces and sampling schemes for which sharp, verifiable approximation guarantees can be obtained, with explicit inverse-type and strong-converse statements. Overall, the paper advances the theoretical foundations of sampling-based approximation by unifying direct, inverse, and strong-converse results via -functionals and Steklov-based smoothness across periodic and non-periodic domains.

Abstract

This paper studies approximation properties of linear sampling operators in general Banach lattices . We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special -functionals and their realizations by sampling operators, as well as strong converse inequalities, which, to the best of our knowledge, have not been previously established for sampling operators even in the classical spaces . The results extend several classical theorems previously known mainly in and apply to all functions for which the corresponding sampling operator is well defined, thereby substantially enlarging the class of functions that can be considered in this framework.
Paper Structure (16 sections, 43 theorems, 198 equations)

This paper contains 16 sections, 43 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Notation and auxiliary results
  3. Banach lattices
  4. Marcinkiewicz–Zygmund (MZ)-type inequalities in Banach lattices
  5. Moduli of smoothness and errors of best approximation
  6. Main results in the periodic case
  7. Estimates in terms of special moduli of smoothness
  8. Strong converse inequalities for sampling operators
  9. Special $K$-functionals and their realizations
  10. Comparison of the errors $\|f-G_n f\|_X$ and $\|f-G_n f\|_{X_n}$
  11. Estimates involving the errors of best and best one-sided approximations
  12. Examples
  13. Main results in the non-periodic case
  14. Estimates in terms of special moduli of smoothness
  15. Estimates in terms of the errors of best and best one-sided approximations
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $X$ be a Banach lattice satisfying Assumption A. If $f \in X$ and $K \in \mathcal{R}^*$, then the convolution $f*K$ exists and is finite almost everywhere, belongs to $X$, and satisfies where the constant $C$ depends only on $X$.

Theorems & Definitions (72)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 62 more