Hardy-Littlewood type inequalities for Laguerre series

Hardy-Littlewood type inequalities for Laguerre series

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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)|rdt≤C∑j=0∞|cj|rj¯1−r/2 for certain r≤1, wh...

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Main Authors: Chin-Cheng Lin, Shu-Huey Lin
Format: Article
Language:English
Published: Hindawi Limited 2002-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/S0161171202108234
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spelling doaj-79177044381c4a10ab687b69ec0353402020-11-24T23:01:34ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0130953354010.1155/S0161171202108234Hardy-Littlewood type inequalities for Laguerre seriesChin-Cheng Lin0Shu-Huey Lin1Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, ChinaDepartment of Mathematics, National Central University, Chung-Li, 320, Taiwan, ChinaLet {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)|rdt≤C∑j=0∞|cj|rj¯1−r/2 for certain r≤1, where f is the limit function of ∑cj𝔏ja. Moreover, we show that if f(x)∼∑cj𝔏ja is in Lr, r≥1, we have the converse Hardy-Littlewood type inequality ∑j=0∞|cj|rj¯β≤C∫0∞|f(t)|rdt for r≥1 and β<−r/2.http://dx.doi.org/10.1155/S0161171202108234
collection DOAJ
language English
format Article
sources DOAJ
author Chin-Cheng Lin
Shu-Huey Lin
spellingShingle Chin-Cheng Lin
Shu-Huey Lin
Hardy-Littlewood type inequalities for Laguerre series
International Journal of Mathematics and Mathematical Sciences
author_facet Chin-Cheng Lin
Shu-Huey Lin
author_sort Chin-Cheng Lin
title Hardy-Littlewood type inequalities for Laguerre series
title_short Hardy-Littlewood type inequalities for Laguerre series
title_full Hardy-Littlewood type inequalities for Laguerre series
title_fullStr Hardy-Littlewood type inequalities for Laguerre series
title_full_unstemmed Hardy-Littlewood type inequalities for Laguerre series
title_sort hardy-littlewood type inequalities for laguerre series
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 2002-01-01
description 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)|rdt≤C∑j=0∞|cj|rj¯1−r/2 for certain r≤1, where f is the limit function of ∑cj𝔏ja. Moreover, we show that if f(x)∼∑cj𝔏ja is in Lr, r≥1, we have the converse Hardy-Littlewood type inequality ∑j=0∞|cj|rj¯β≤C∫0∞|f(t)|rdt for r≥1 and β<−r/2.
url http://dx.doi.org/10.1155/S0161171202108234
work_keys_str_mv AT chinchenglin hardylittlewoodtypeinequalitiesforlaguerreseries
AT shuhueylin hardylittlewoodtypeinequalitiesforlaguerreseries
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