Volume 25, pp. 467-479, 2006.

On convergence of orthonormal expansions for exponential weights

H. P. Mashele

Abstract

Let $I=(-d,d)$ be a real interval, finite or infinite, and let $W:I\to (0,\mathrm{\infty })$. Assume that $W^2$ is a weight, so that we may define orthonormal polynomials corresponding to $W^2$. For $f:I\to \mathbf{R}$, let $s_m\left[f\right]$ denote the $m$th partial sum of the orthonormal expansion of $f$ with respect to these polynomials. We show that if $f^{\prime }W\in L_{\mathrm{\infty }}\left(I\right)\cap L_2\left(I\right)$, then ${\Vert (s_m\left[f\right]-f)W\Vert }_{L_{\mathrm{\infty }}\left(I\right)}\to 0$ as $m\to \mathrm{\infty }$. The class of weights considered includes even exponential weights.

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Key words

orthonormal polynomials, de la Vallée Poussin means

AMS subject classifications

65N12, 65F35, 65J20, 65N55