Approximation by linear sampling operators in Banach spaces

TL;DR

This paper investigates the approximation capabilities of linear sampling operators within Banach lattices, establishing new theoretical results that extend classical theorems and apply to a broader class of functions.

Contribution

It provides the first comprehensive set of direct and inverse approximation estimates, convergence criteria, and inequalities for sampling operators in Banach spaces, expanding their applicability.

Findings

Established matching approximation estimates and convergence criteria.

Proved equivalence results involving $K$-functionals and sampling operators.

Extended classical theorems to a wider class of functions in Banach lattices.

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.