We investigate approximation properties of Cesàro (C; −α, −β)-means of double Walsh-Fourier series with α, β ∈ (0, 1). As an application, we obtain a sufficient condition for the convergence of the means σ
(f; x, y) to f(x,y) in the Lp-metric, p ∈ [1, ∞]. We also show that this sufficient condition cannot be improved.
The main aim of this paper is to prove that the maximal operator σ*α of the (C, α) means of the cubical partial sums of the two-dimensional Walsh-Fourier series is bounded from the Hardy space H2/(2+α) to the space weak-L2/(2+α).
The main aim of this paper is to prove that there exists a martingale f ∈ H12/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier
series does not belong to the space weak-L1/2.
The main aim of this paper is to prove that the maximal operator σ0k*:= supn ∣σn,nk∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H1/2 to the space weak-L1/2.
In this paper we study the exponential uniform strong approximation of two-dimensional Walsh-Fourier series. In particular, it is proved that the two-dimensional Walsh-Fourier series of the 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 best possible.