Year: 1990
Type: article
Abstract: We apply a construction of generalized convolution in \[ {L^1}({\mathbb {R}_ + } \times \mathbb {R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,\] 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 \[ L = - \left ( {\frac {{{\partial ^2}}} {{\partial {x^2}}} + \frac {{2\alpha - 1}} {x}\frac {\partial } {{\partial x}} + {x^2}\frac {{{\partial ^2}}} {{\partial {t^2}}}} \right ),\qquad x > 0,\quad t \in \mathbb {R},\quad \alpha \geqslant 1.\] 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.
Author: Krzysztof Stempak
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2010519560