Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in Lp(Q)Lp(Q) with 1≤p≤∞1≤p≤∞. Here QQ is a dd-parallelepiped in RdRd with sides parallel to the coordinate axes. We consider the error of best approximation of a function ff by algebraic polynomials of fixed degree at most ri−1ri−1 in variable xi,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.

► We proved a multivariate Whitney’s theorem for an anisotropic polynomial approximation. ► Functions to be approximated are defined on a parallelepiped. ► The degrees of polynomials for approximation are fixed but different in each variable. ► Suppose that the size of parallelepiped is going to zero. ► Then the rate of approximation error is estimated by a total mixed modulus of smoothness.