Uniform approximation by multivariate quasi-projection operators

TL;DR

This paper investigates the approximation capabilities of multivariate quasi-projection operators, providing uniform error estimates and conditions for optimal approximation in various function spaces.

Contribution

It introduces new error bounds for quasi-projection operators associated with functions satisfying Strang-Fix conditions, extending approximation theory in multivariate settings.

Findings

Established uniform norm error estimates for a broad class of quasi-projection operators.

Derived two-sided approximation bounds using K-functional realizations.

Extended approximation results to anisotropic Besov spaces.

Abstract

Approximation properties of quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$ are studied. Such an operator is associated with a function $\varphi$ satisfying the Strang-Fix conditions and a tempered distribution $\widetilde{\varphi}$ such that compatibility conditions with $\varphi$ hold. Error estimates in the uniform norm are obtained for a wide class of quasi-projection operators defined on the space of uniformly continuous functions and on the anisotropic Besov spaces. Under additional assumptions on $\varphi$ and $\widetilde{\varphi}$, two-sided estimates in terms of realizations of the $K$-functional are also obtained.