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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Mean Cesàro summability of Laguerre and Hermite series

Author: Eileen L. Poiani
Journal: Trans. Amer. Math. Soc. 173 (1972), 1-31
MSC: Primary 42A56; Secondary 33A65
MathSciNet review: 0310537
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Abstract: The primary purpose of this paper is to prove inequalities of the type $ \vert\vert{\sigma _n}(f,x)W(x)\vert{\vert _p} \leqslant C\vert\vert f(x)W(x)\vert{\vert _p}$ where $ {\sigma _n}(f,x)$ is the $ n$th $ (C,1)$ mean of the Laguerre or Hermite series of $ f, W(x)$ is a suitable weight function of particular form, $ C$ is a constant independent of $ f(x)$ and $ n$, and the norm is taken over $ (0,\infty )$ in the Laguerre case and $ ( - \infty ,\infty )$ in the Hermite case for $ 1 \leqslant p \leqslant \infty $. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For $ p < \infty $ and $ f(x)W(x) \in {L^p}$, mean summability results showing that $ \mathop {\lim }\nolimits_{n \to \infty } \vert\vert[{\sigma _n}(f,x) - f(x)]W(x)\vert{\vert _p} = 0$ are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.

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Keywords: Mean Cesàro summability, Laguerre series, Hermite series, norm inequalities, weight function, Cesàro kernel estimates
Article copyright: © Copyright 1972 American Mathematical Society