Whitney's theorem for local anisotropic polynomial L_p-approximation, 0<p<1

Abstract

Dinh Dũng and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in $L_p(Q)$ for $1\le p\le \infty$, where $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. They considered the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i-1$ in variable $x_i,i=1,...,d$. The convergence rate of the approximation error when the size of $Q$ going to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable to the case $0<p<1$. In the present paper, by a different method we proved this theorem for $0<p\le \infty$.

arXiv commentarXiv admin note: text overlap with arXiv:1007.1362 by other authors