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\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}
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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}

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