On almost everywhere and mean convergence of Hermite and Laguerre expansions

Thangavelu, S. (1990) On almost everywhere and mean convergence of Hermite and Laguerre expansions Colloquium Mathematicum, 60 . pp. 21-34. ISSN 0010-1354

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Official URL: http://journals.impan.gov.pl/cgi-bin/shvold?cm60

Abstract

Namely, for 1 ≤ p ≤ ∞, α > (n − 1)/2 the boundedness of the operator Sα R : Lp → Lp is proved, where SαRƒ are Riesz means of order α of Hermite expansions of a function ƒ (cf. loc.cit.). The a.e. convergence of SαRƒ(x) to ƒ(x) for ƒ ∈ Lp (Rp), p ≥ 2, n ≥ 2 and α > (n − 1)(1/2 − 1/p) and the convergence of SαRƒ(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 σαnƒ(r) of order α > (2n − 1)(1/2 − 1/p)of Laguerre expansions of a function ƒ ∈ Lp (R+, r2n−1dr), 2 ≤ p ≤ ∞ (the notations are from the author.

Item Type:Article
Source:Copyright of this article belongs to Institute of Mathematics of the Polish Academy of Sciences.
Keywords:Laguerre Expansion; Riesz Means; Hermite expansion; Lebesgue point; a.e.Convergence
ID Code:64415
Deposited On:10 Oct 2011 05:46
Last Modified:29 Nov 2011 11:02

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