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L 1-convergence of double cosine- and Walsh-Fourier series

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Abstract

We prove the convergence inL 1([−gp, π)2)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa jk satisfying certain conditions of Hardy-Karamata kind, and such thata jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL 1 (0, 1)2)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720, Szeged, Hungary

    Ferenc Móricz

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  1. Ferenc Móricz
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Additional information

This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.

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Móricz, F. L 1-convergence of double cosine- and Walsh-Fourier series. J. Anal. Math. 62, 115–130 (1994). https://doi.org/10.1007/BF02835950

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  • DOI: https://doi.org/10.1007/BF02835950

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