On Whitney type inequalities for local anisotropic polynomial approximation

2010/07/08 by Doan Cong Dinh, Tino Ullrich, Dinh, D. +1
Mathematics · #Mathematical Approximation and Integration #Mathematical functions and polynomials #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.1007.1362

Abstract

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤ p≤ ∞. 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.

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