Mean summability methods for Laguerre series

Abstract

We apply a construction of generalized convolution in \[ L 1 ( R + × R , x 2 α − 1 d x d t ) , α ⩾ 1 , {L^1}({\mathbb {R}_ + } \times \mathbb {R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1, \] 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 \[ L = − ( ∂ 2 ∂ x 2 + 2 α − 1 x ∂ ∂ x + x 2 ∂ 2 ∂ t 2 ) , x > 0 , t ∈ R , α ⩾ 1. L = - \left ( {\frac {{{\partial ^2}}} {{\partial {x^2}}} + \frac {{2\alpha - 1}} {x}\frac {\partial } {{\partial x}} + {x^2}\frac {{{\partial ^2}}} {{\partial {t^2}}}} \right ),\qquad x > 0,\quad t \in \mathbb {R},\quad \alpha \geqslant 1. \] 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.