On Whitney type inequalities for local anisotropic polynomial approximation

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

Whitney Multiapproximation

برهنا في هذا البحث نظرية  وتني  لأفضل تقريب متعدد للدالة  التي تنتمي إلى الفضاء  بواسطة متعددة الحدود الجبرية متعددة المتغيرات  من الدرجة اقل أو تساوي  .In this article we  prove  that Whitney  theorem  for  the  value of the  best  multiapproximation of a function   by algebraic multipolynomial  of degree