Whitney type inequalities for local anisotropic polynomial approximation Original Research Article

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in image with image. Here image is a image-parallelepiped in image with sides parallel to the coordinate axes. We consider the error of best approximation of a function image by algebraic polynomials of fixed degree at most image in variable image, 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.