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Transactions of the American Mathematical Society

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



Mean summability methods for Laguerre series

Author: Krzysztof Stempak
Journal: Trans. Amer. Math. Soc. 322 (1990), 671-690
MSC: Primary 42C10; Secondary 43A55
MathSciNet review: 974528
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Abstract: We apply a construction of generalized convolution in

$\displaystyle {L^1}({\mathbb{R}_ + } \times \mathbb{R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,$

cf. [8], to investigate the mean convergence of expansions in Laguerre series. Following ideas of [4, 5] we construct a functional calculus for the operator

$\displaystyle L = - \left( {\frac{{{\partial ^2}}} {{\partial {x^2}}} + \frac{{... ... {t^2}}}} \right),\qquad x > 0,\quad t \in \mathbb{R},\quad \alpha \geqslant 1.$

Then, arguing as in [3], we prove results concerning the mean convergence of some summability methods for Laguerre series. In particular, the classical Abel-Poisson and Bochner-Riesz summability methods are included.

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Keywords: Laguerre expansions, generalized convolution, mean convergence
Article copyright: © Copyright 1990 American Mathematical Society

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