Mean summability methods for Laguerre series. (English) Zbl 0713.42024

Askey and Wagner settled the question of mean convergence of Laguerre expansions in the sense that mean convergence takes place in \(L^ p\) if and only if \(4/3<p<4.\) Then Muckenhoupt proved mean convergence of Abel means in every \(L^ p,1\leq p\leq \infty.\) More general summation methods were investigated by Dlugosz in the case when the Laguerre parameter is an integer. In the present work this restriction is dropped, a norm summability result is proved where the summability kernel is quite general, e.g. it may be taken to be \(K(\lambda)=\exp (-\lambda)\) or \(K(\lambda)=(1-\lambda)^ N_+.\) The method is based on a generalized convolution in \(L^ 1({\mathbb{R}}_+\times {\mathbb{R}},x^{2\alpha -1} dx dt).\) In the process a functional calculus is developed for the operator \[ L=-(\frac{\partial^ 2}{\partial x^ 2}+\frac{2\alpha -1}{x} \frac{\partial}{\partial x}+x^ 2\frac{\partial^ 2}{\partial t^ 2}),\quad x>0,\quad t\in {\mathbb{R}},\quad \alpha \geq 1. \]
Reviewer: V.Totik


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C15 General harmonic expansions, frames
43A55 Summability methods on groups, semigroups, etc.
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