Convergence in measure of strong logarithmic means of double Fourier series

Nörlund strong logarithmic means of double Fourier series acting from space L log L \(\left( {\mathbb{T}^2 } \right)\) into space L _p \(\left( {\mathbb{T}^2 } \right)\) , 0 < p < 1, are studied. The maximal Orlicz space such that the Nörlund strong logarithmic means of double Fourier series for the functions from this space converge in two-dimensional measure is found.     

On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero

The aim of this paper is to prove that for an arbitrary set of measure zero there exists a bounded function for which the Fejér means of the Walsh-Fourier series of the function diverge. Research supported by the Georgian National Fundation for Scientific Research, Grant no. 07_225_3-100.     

Negative order Cesàro means of double Fourier series and bounded generalized variation

Under study is the convergence of the negative order Cesàro means of the double trigonometric Fourier series of functions of bounded generalized Λ-variation.     

Strong approximation by Marcinkiewicz means of two-dimensional Walsh–Kaczmarz–Fourier series

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series of every continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is the best possible.     

On the convergence of multiple Fourier series of functions of bounded partial generalized variation

The convergence of multiple Fourier series of functions of bounded partial Λ-variation is investigated. The sufficient and necessary conditions on the sequence Λ = {λ _n } are found for the convergence of multiple Fourier series of functions of bounded partial Λ-variation.     

Strong summability theorems for H p (Δ d )

We prove that certain means of the (C,α,…,α)-means (α=1/p?1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space H _p to H _p (1≦p≦2). As a consequence we obtain strong summability theorems concerning (C,α,…,α)-means.     

Strong Summability of Two-Dimensional Vilenkin–Fourier Series

Ukrainian Mathematical Journal - We study the exponential uniform strong summability of two-dimensional Vilenkin–Fourier series. In particular, it is proved that the two-dimensional...     

BMO-Estimation and Almost Everywhere Exponential Summability of Quadratic Partial Sums of Double Fourier Series

A \(\mathrm{BMO\,}\) -estimation for quadratic partial sums of two-dimensional Fourier series is proved from which is derived an almost everywhere exponential summability of quadratic partial sums of double Fourier series.     

On the convergence of multiple Walsh-Fourier series of functions of bounded generalized variation

The convergence of multiple Walsh-Fourier series of functions of bounded generalized variation is investigated. The sufficient and necessary conditions on the sequence ?? = {?? _n } are found for the convergence of multiple Walsh-Fourier series of functions of bounded partial ??-variation.