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Proceedings of the American Mathematical Society

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Convergence and integrability of double trigonometric series with coefficients of bounded variation


Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 102 (1988), 633-640
MSC: Primary 42B05
DOI: https://doi.org/10.1090/S0002-9939-1988-0928995-2
MathSciNet review: 928995
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Abstract: We prove that if $ c\left( {j,k} \right) \to 0$ as $ \max \left( {\vert j\vert,\vert k\vert} \right) \to \infty $ and

$\displaystyle \sum\limits_{j = - \infty }^\infty {\sum\limits_{k = - \infty }^\infty {\left\vert {{\Delta _{11}}c\left( {j,k} \right)} \right\vert < \infty ,} } $

then the series $ \sum\nolimits_{j = - \infty }^\infty {\sum\nolimits_{k = - \infty }^\infty {c\left( {j,k} \right){e^{i(jx + ky)}}} } $ converges both pointwise for every $ \left( {x,y} \right) \in {\left( {T\backslash \left\{ 0 \right\}} \right)^2}$ and in the $ {L^p}\left( {{T^2}} \right)$-metric for $ 0 < p < 1$, where $ T$ is the one-dimensional torus. Both convergence statements remain valid for the three conjugate series under these same coefficient conditions.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928995-2
Keywords: Double trigonometric series, conjugate series, double sequences of bounded variation, pointwise convergence, convergence in the $ {L^p}\left( {{T^2}} \right)$-metric for $ 0 < p < 1$
Article copyright: © Copyright 1988 American Mathematical Society

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