Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 32, No. 2, pp. 215-223 (2016)

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

György Gát

University of Debrecen

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 the theorem of Carleson for the Walsh-Paley system, that is, for each function in $L^2$ we have the almost everywhere convergence $S_nf\to f$ and Fine proved the Fejér-Lebesgue theorem, that is for each integrable function in $L^1$ we have the almost everywhere convergence of Fejér means $\sigma_nf\to f$. In 1992 Moricz, Schipp, and Wade proved that for each two-variable function in the space $L\log^+L$ the Fejér 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.

Keywords: Walsh-Paley system, one and two-dimensional Fejér means, logarithmic means, maximal convergence spaces, subsequence of partial sums, Marcinkiewicz-like means.

Classification (MSC2000): 42C10

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