On almost everywhere and mean convergence of Hermite and Laguerre expansions

Abstract

Namely, for 1 ≤ p ≤ ∞, α > (n − 1)/2 the boundedness of the operator S α R : L p → L p is proved, where S α R ƒ are Riesz means of order α of Hermite expansions of a function ƒ (cf. loc.cit.). The a.e. convergence of S α R ƒ(x) to ƒ(x) for ƒ ∈ L p (R p ), p ≥ 2, n ≥ 2 and α > (n − 1)(1/2 − 1/p) and the convergence of S α R ƒ(x) to ƒ(x) at every Lebesgue point x of ƒ if α > (n − 1)/2 are proved. Moreover the a.e. convergence of Riesz means σ α n ƒ(r) of order α > (2n − 1)(1/2 − 1/p)of Laguerre expansions of a function ƒ ∈ L p (R + , r 2n−1 dr), 2 ≤ p ≤ ∞ (the notations are from the author.