Approximation by multivariate quasi-projection operators and Fourier multipliers

TL;DR

This paper investigates the approximation capabilities of multivariate quasi-projection operators using Fourier multipliers, providing error estimates for functions in Besov and Lp spaces.

Contribution

It introduces new approximation results for quasi-projection operators satisfying Strang-Fix and compatibility conditions, with detailed error bounds.

Findings

Derived upper and lower bounds for approximation errors

Established estimates in terms of moduli of smoothness

Extended analysis to anisotropic Besov and Lp spaces

Abstract

Multivariate quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$, associated with a function $\varphi$ and a distribution/function $\widetilde{\varphi}$, are considered. The function $\varphi$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\widetilde{\varphi}$. Using technique based on the Fourier multipliers, we studied approximation properties of such operators for functions $f$ from anisotropic Besov spaces and $L_p$ spaces with $1\le p\le \infty$. In particular, upper and lower estimates of the $L_p$-error of approximation in terms of moduli of smoothness and best approximations are obtained.