Some Recent Results on Convergence and Divergence with Respect to Walsh-fourier Series

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 we have the almost everywhere convergence Snf → f and Fine proved [4] the Fejér-Lebesgue theorem, that is for each integrable function in L we have the almost everywhere convergence of Fejér means σnf → f . In 1992 Móricz, Schipp and Wade proved [18], 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. Introduction Let the numbers n ∈ N and x ∈ I := [0, 1) be expanded with respect to the binary number system: