Volume 25, pp. 467-479, 2006.

On convergence of orthonormal expansions for exponential weights

H. P. Mashele


Let $I=\left( -d,d\right) $ be a real interval, finite or infinite, and let $% W:I\rightarrow \left( 0,\infty \right) $. Assume that $W^{2}$ is a weight, so that we may define orthonormal polynomials corresponding to $W^{2}$. For $% f:I\rightarrow {\bf R}$, 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_{\infty }\left( I\right) \cap L_{2}\left( I\right) $, then $\left\Vert \left( s_{m}\left[ f\right] -f\right) W\right\Vert _{L_{\infty }\left( I\right) }\rightarrow 0$ as $% m\rightarrow \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

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