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Weighted Integrability of Double Trigonometric Series

Proceedings of the American Mathematical Society / v.127 no.5. 1999, pp.1463-1471

Author : Chen, Chang-Pao ; Huang, Xin-Rong

Abstract : We study the double trigonometric series whose coefficients c $_{jk}$ are such that $\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty} c_{jk} . Then its rectangular partial sums converge uniformly to some f $\in $ C(T $^{2}$ ). We give sufficient conditions for the Lebesgue integrability of {f(x, y) - f(x, 0) - f(0, y) + f(0, 0)} $\phi $ (x, y), where $\phi $ (x, y) = 1/xy, 1/x, or 1/y. For certain cases, they are also necessary conditions. Our results extend those of Boas and Moricz from the one-dimensional to the two-dimensional series.

Keyword : Primary 42B99, 42A16

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