Mean Cesàro summability of Laguerre and Hermite series

Abstract:

The primary purpose of this paper is to prove inequalities of the type $||{\sigma _n}(f,x)W(x)|{|_p} \leqslant C||f(x)W(x)|{|_p}$ 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 $\lim \nolimits _{n \to \infty } ||[{\sigma _n}(f,x) - f(x)]W(x)|{|_p} = 0$ 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.