On Whitney type inequalities for local anisotropic polynomial approximation

Abstract

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in $L_p(Q)$ with $1\le p\le \infty$. 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.

12 pages; the proofs of Theorems 1.2 and 2.2 and Lemma 3.1 are revised; typos are corrected; Acknowledgments are added; the results are unchanged