International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2002 / Article

Open Access

Volume 30 |Article ID 736831 | 8 pages | https://doi.org/10.1155/S0161171202108234

Hardy-Littlewood type inequalities for Laguerre series

Received30 Aug 2001

Abstract

Let {cj} be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on {cj} to obtain the pointwise convergence as well as Lr-convergence of Laguerre series cj𝔏ja. Then, we prove a Hardy-Littlewood type inequality 0|f(t)|rdtCj=0|cj|rj¯1r/2 for certain r1, where f is the limit function of cj𝔏ja. Moreover, we show that if f(x)cj𝔏ja is in Lr, r1, we have the converse Hardy-Littlewood type inequality j=0|cj|rj¯βC0|f(t)|rdt for r1 and β<r/2.

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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