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.
Copyright
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.
PDF
