scholarly journals L1-convergence of double trigonometric series

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3759-3771
Author(s):  
Karanvir Singh ◽  
Kanak Modi
Keyword(s):  
Trigonometric Series ◽  
Bounded Variation ◽  
Pointwise Convergence ◽  
Null Sequence ◽  
Sine Series ◽  
Double Trigonometric Series ◽  
L1 Norm ◽  
Cosine Series

In this paper we study the pointwise convergence and convergence in L1-norm of double trigonometric series whose coefficients form a null sequence of bounded variation of order (p,0),(0,p) and (p,p) with the weight (jk)p-1 for some integer p > 1. The double trigonometric series in this paper represents double cosine series, double sine series and double cosine sine series. Our results extend the results of Young [9], Kolmogorov [4] in the sense of single trigonometric series to double trigonometric series and of M?ricz [6,7] in the sense of higher values of p.

scholarly journals Double trigonometric series with coefficients of bounded variation of higher order

2004 ◽  
Vol 35 (3) ◽  
pp. 267-280 ◽  
Author(s):  
Kulwinder Kaur ◽  
S. S. Bhatia ◽  
Babu Ram
Keyword(s):  
Uniform Convergence ◽  
Trigonometric Series ◽  
Bounded Variation ◽  
Pointwise Convergence ◽  
Higher Order ◽  
Null Sequence ◽  
Partial Sums ◽  
Convergence Properties ◽  
Double Trigonometric Series

In this paper the following convergence properties are established for the rectangular partial sums of the double trigonometric series, whose coefficients form a null sequence of bounded variation of order $ (p,0) $, $ (0,p) $ and $ (p,p) $, for some $ p\ge 1$: (a) pointwise convergence; (b) uniform convergence; (c) $ L^r $-integrability and $ L^r $-metric convergence for $ 0

scholarly journals Hardy-Littlewood type inequalities for Laguerre series

2002 ◽  
Vol 30 (9) ◽  
pp. 533-540
Author(s):  
Chin-Cheng Lin ◽  
Shu-Huey Lin
Keyword(s):  
Bounded Variation ◽  
Pointwise Convergence ◽  
Type Inequality ◽  
Growth Conditions ◽  
Null Sequence ◽  
Limit Function ◽  
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 asLr-convergence of Laguerre series∑cj𝔏ja. Then, we prove a Hardy-Littlewood type inequality∫0∞|f(t)|rdt≤C∑j=0∞|cj|rj¯1−r/2for certainr≤1, wherefis the limit function of∑cj𝔏ja. Moreover, we show that iff(x)∼∑cj𝔏jais inLr,r≥1, we have the converse Hardy-Littlewood type inequality∑j=0∞|cj|rj¯β≤C∫0∞|f(t)|rdtforr≥1andβ<−r/2.

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