Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series

Abstract

In this article we discuss the Nörlund means of cubical partial sums of Walsh-Fourier series of a function in $L^p$ ($1\leq p\leq \infty$). We investigate the rate of the approximation by this means, in particular, in $\textrm {Lip}(\alpha,p),$ where $\alpha>0$ and $1\leq p\leq\infty$. In case $p=\infty$ by $L^p$ we mean $C_W$, the collection of the uniformly $W$-continuous functions. Our main theorems state that the approximation behavior of the two-dimensional Walsh-Nörlund means is so good as the approximation behavior of the one-dimensional Walsh-Nörlund means.

As special case we get the Nörlund logarithmic means of cubical partial sums of Walsh-Fourier series discussed recently by Gát and Goginava in 2004 [5] and the $(C,\beta)$-means of Marcinkiewicz type with respect to double Walsh-Fourier series discussed by Goginava [10].

Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series was given by Móricz and Siddiqi [14].