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Almost everywhere strong summability of double Walsh-Fourier series

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Abstract

In this paper we study a question of almost everywhere strong convergence of the quadratic partial sums of two-dimensional Walsh-Fourier series. Specifically, we prove that the asymptotic relation \(\tfrac{1} {n}\sum\limits_{m = 0}^{n - 1} {|S_{mm} f - f|^p \to 0} \) as n→∞ holds a.e. for every function of two variables belonging to L logL and for 0 < p ≤ 2. Then using a theorem by Getsadze [6] we infer that the space L log L can not be enlarged by preserving this strong summability property.

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Correspondence to G. Gát.

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Original Russian Text © G. Gát, U. Goginava, 2015, published in Izvestiya NAN Armenii. Matematika, 2015, No. 1, pp. 23–40.

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Gát, G., Goginava, U. Almost everywhere strong summability of double Walsh-Fourier series. J. Contemp. Mathemat. Anal. 50, 1–13 (2015). https://doi.org/10.3103/S106836231501001X

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  • DOI: https://doi.org/10.3103/S106836231501001X

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