Universality properties of double series by generalized Walsh system

Abstract

In this paper we consider a question on existence of double series by generalized Walsh system, which are universal in weighted $L_{\mu }^1[0,1{]}^2$ spaces. In particular, we construct a weighted function $\mu (x,y)$ and a double series by generalized Walsh system of the form $$\sum_{n,k=1}^\infty c_{n,k}ψ_n(x)ψ_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left | c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$ which is universal in $L_{\mu }^1[0,1{]}^2$ concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.