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On double cosine, sine, and Walsh series with monotone coefficients

Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 109 (1990), 417-425
MSC: Primary 42B05; Secondary 42A32
MathSciNet review: 1010803
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Abstract: We extend from one-dimensional to two-dimensional series the results by Hardy and Littlewood [6] on the $ {L^r}$-integrability of the sum $ f$ and the results by Stechkin [10] on the $ {L^1}$-integrability of the maximum partial sum $ {M^*}$ in the case of cosine and sine series with monotone coefficients. Among others, we prove that the $ {L^r}$-integrability of $ f$ and $ {M^*}$ is essentially equivalent for $ r > 1$ in the two-dimensional setting, too. Simultaneously, we extend our earlier results in [7] from one-dimensional to two-dimensional Walsh series.

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Keywords: Rectangular partial sum, maximum partial sum, pointwise convergence and convergence in $ {L^r}$-norm in Pringsheim's sense, Hardy's inequalities for double integrals and double sequences
Article copyright: © Copyright 1990 American Mathematical Society

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