Approximation by multivariate quasi-projection operators and Fourier multipliers

Abstract

Multivariate quasi-projection operators $Q_j(f,\phi ,\tilde{\phi })$, associated with a function $\phi$ and a distribution/function $\tilde{\phi }$, are considered. The function $\phi$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\tilde{\phi }$. 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.