Existence of double Walsh series universal in weighted $L_\mu^1[0,1]^2$ spaces

TL;DR

This paper constructs a weighted function and a double Walsh series that is universal in weighted $L^1$ spaces on the unit square, capable of approximating functions via subseries with convergence in both spherical and rectangular partial sums.

Contribution

It demonstrates the existence of a double Walsh series that is universal in weighted $L^1$ spaces, a novel result in the theory of orthogonal series.

Findings

Constructed a specific weighted function $ul(x,y)$

Developed a double Walsh series with coefficients satisfying $or all q>2$

Proved universality concerning convergence of subseries in $L_ul^1$ spaces

Abstract

In this paper we consider a question on existence of double Walsh series universal in weighted $L_\mu^1[0,1]^2$ spaces. We construct a weighted function $\mu(x,y)$ and a series by double Walsh system of the form $$\sum_{n,k=1}^\infty c_{n,k}W_n(x)W_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left | c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$ 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.