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.     

On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

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...     

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.     

APPROXIMATION PROPERTIES OF （C, α） MEANS OF DOUBLE WALSH-FOURIER SERIES

We study the rate of Lp approximation by Cesaro means of the quadratic partial sums of double Walsh-Fourier series of functions from Lp.     

On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)$\Phi (t):[0,\infty )\to [0,\infty )$ is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.