On Whitney type inequalities for local anisotropic polynomial approximation

TL;DR

This paper establishes a multivariate Whitney type theorem for local anisotropic polynomial approximation in $L_p$ spaces, relating approximation error to a mixed modulus of smoothness, and introduces a new equivalence theorem involving K-functionals.

Contribution

It provides the first Whitney type theorem for anisotropic polynomial approximation in multivariate $L_p$ spaces, linking approximation error to a total mixed modulus of smoothness.

Findings

Proved a multivariate Whitney type theorem for anisotropic polynomial approximation.

Established an equivalence between a K-functional and the total mixed modulus of smoothness.

Derived convergence rate characterizations for polynomial approximation in $L_p$ spaces.

Abstract

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in $L_p(Q)$ with $1\leq p\leq \infty$. Here $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. We consider the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in variable $x_i,\ i=1,...,d$, and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.