Order summability of multiple Fourier series

Peterson, G. E.; Welland, G. V.

doi:10.1090/S0002-9947-1975-0374807-3

Order summability of multiple Fourier series
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by G. E. Peterson and G. V. Welland PDF

Trans. Amer. Math. Soc. 205 (1975), 221-246 Request permission

Abstract:

Jurkat and Peyerimhoff have characterized monotone Fouriereffective summability methods as those which are stronger than logarithmic order summability. Here the analogous result for double Fourier series is obtained assuming unrestricted rectangular convergence. It is also shown that there is a class of order summability methods, which are weaker than any Cesàro method, for which the double Fourier series of any $f \in L$ is restrictedly summable almost everywhere. Finally, it is shown that square logarithmic order summability has the localization property for exponentially integrable functions.

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On the summability of double Fourier series

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