György Gát And

Restricted convergence of Fejér means of trigonometric Fourier series

Convergence and Divergence of Fejér Means of Fourier Series on One and Two-Dimensional Walsh and Vilenkin Groups

It is a highly celebrated issue in dyadic harmonic analysis the pointwise convergence of the Fejér (or (C,1)) means of functions on the Walsh and Vilenkin groups both in the point of view of one and two dimensional cases. We give a résumé of the very recent developments concerning this matter, propose unsolved problems and throw a glance at the investigation of Vilenkin-like systems too.

Almost Everywhere Convergence of a Subsequence of the Logarithmic Means of Vilenkin-Fourier Series

Abstract: The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1,1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1,1), provided that the supremum in the maximal operator is tak...

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 funct...

Convergence of Cesàro Means of Functions with Respect to Unbounded Vilenkin Systems

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. The aim of this paper is to give a résumé of the recent developments concerning this matter. Above all, we prove that the maximal operator sup |σMn | is of type (H, L1) on unbounded Vilenkin groups. First, we give a brief introduc...