Some recent results on convergence and divergence with respect to Walsh-Fourier series

Gát, György (2016) Some recent results on convergence and divergence with respect to Walsh-Fourier series. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 32 (2). pp. 215-223. ISSN 0866-0174

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Abstract

It is of main interest in the theory of Fourier series the reconstruction of a function from the partial sums of its Fourier series. Just to mention two examples: Billard proved [2] the theorem of Carleson for the Walsh-Paley system, that is, for each function in L 2 we have the almost everywhere convergence Snf → f and Fine proved [4] the Fej´er-Lebesgue theorem, that is for each integrable function in L 1 we have the almost everywhere convergence of Fej´er means σnf → f. In 1992 M´oricz, Schipp and Wade proved [18], that for each two-variable function in the space Llog+ L the Fej´er means of the two-dimensional Walsh-Fourier series converge to the function almost everywhere. In this paper we summarize some results with respect to this issue concerning convergence and also divergence.

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