\documentclass[12pt,thmsa]{article} \usepackage{amsfonts} \usepackage{sw20bams} \input tcilatex \begin{document} \begin{center} \textbf{Polynomial Approximation of Functions in Weighted Sobolev Space} \textit{H.\ ATAMNI}$^1$\textit{\ and M. EL HATRI}$^2$ $\left( 1\right)$\textit{\ Fac. Sci. Dhar El Mahrez, F\{e}s, Morocco} $\left( 2\right)$\textit{\ Laboratoire de Calcul Scientifique, BP 2427 Fes, Morocco} \smallskip\ \textbf{The Abstract} \end{center} The paper deals with the polynomial approximation of functions defined on domain $\Omega$ of $\Bbb{R}^n,n\geq 1$ and belong to Kufner's weighted Sobolev space$^3$ $W^{m,p}(\Omega ,\sigma ).$ The obtained approximation in this paper is a generalization of the results of El Hatri$^2$, and it represents an extension of that of T. Dupont and R. Scott$^1$. This result can be used to estimate the error of Finite Element Method for degenerated and singular Boundary Value Problemes. In this way, we establish the following important result THEOREM.\textbf{\ }\textit{Let }$\left\{ P_j\right\} _{j=1}^k$\textit{\ be a set of nontrivial homogeneous polynomials (in }$\mathit{n}$\textbf{\ }% \textit{variables) of degree}\textbf{\ }$l_j\ ,j=1,2,..k,\$\textit{% respectively; having no common (nonzero) complex zero (this forces }$k\geq n).\$\textit{Define }$\mathcal{K}\Bbb{=}\left\{ v\in C_0^\infty \left( \Bbb{% R}^n\right) :P_j\left( \partial \right) v=0,j=1,...,k\right\} .$ \textit{% Then }$\mathcal{K}\subset P_\kappa ,$ \textit{where }$P_\kappa$\textit{\ is a class of polynomials of degree }$\leq \kappa ,$\textit{\ for some integer} $\kappa$.\textit{\ } \textit{Let }$l=\stackunder{1\leq j\leq k}{\min }l_j,\ m\$\textit{be a nonnegative integer less than }$\mathit{l}\ ,$\textit{\ and let }$1\leq q\leq \infty ,$\textit{\ }$1\leq p_j\leq \infty ,\dfrac 1{r_j}=1-\dfrac 1{p_j}+\dfrac 1q,\ r_j\leq q<\infty .$ \textit{Then, if the following conditions } $r_j(l-m-\theta -n)+\mu q+n\ \rangle 0\quad for\quad 1\leq q\leq \infty ,\theta \in \Bbb{R}$ \textit{hold}$,$ \textit{there is a constant }$c=c(\Omega )$\textit{\ such that} $\stackunder{Q\in \mathcal{K}}{\inf }\Vert f-Q\Vert _{W^{m,q}(\Omega ,\mu )}\leq c(\Omega )\stackunder{j=1}{\dsum^k}\Vert P_j(\partial )f\Vert _{L_{p_j}(\Omega ,\eta _j)},\mathit{\ }\eta _j=\theta +\mu \left( 1-\dfrac q{r_j}\right)$ ACKNOWLEDGMENT:\textbf{\ }\textit{The present paper is supported by the interuniversity cooperation between Morocco and France n}$^{\circ }$\textit{% \ 93/637.} \begin{thebibliography}{9} \bibitem{1} \ J. Dupont, R. Scott\textrm{\ }''Polynomial approximation of functions in Sobolev space'' Math. Comp 34 (1980)150 \bibitem{2} \ M. El\ Hatri$\mathcal{\ }$''Estimation optimale et de type superconvergence de l'erreur de la m\'{e}thode des \'{e}l\'{e}ments finis pour un probl\{e}me aux limites d\'{e}g\'{e}n\'{e}r\'{e}'' RAIRO M$^2$AN, v.21 n$^{\circ }$ 1, 1987, p27-62 \bibitem{3} \ A. Kufner$\mathrm{\ }$''Weighted Sobolev space'', Ed John Willey and sons, 1985. \end{thebibliography} \end{document} 