Hardy-Littlewood type inequalities for Laguerre series
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202108234 |
| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |