Almost everywhere convergence of Fejér means of two-dimensional triangular Walsh-Fourier series

Abstract

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence ${\sigma }_n^{△}f=\frac{1}{n}{\sum }_{k=0}^{n-1}{S}_{k,n-k}f\to f$ ($n\to \infty$) for each integrable two-variable function $f$.