Universality properties of double series by generalized Walsh system

TL;DR

This paper constructs a weighted double series using a generalized Walsh system that is universal in weighted L^1 spaces, capable of approximating functions through subseries with convergence in both spherical and rectangular partial sums.

Contribution

It introduces a new universal double series in weighted L^1 spaces based on a generalized Walsh system, with specific convergence properties.

Findings

Constructed a weighted function μ(x,y) for universality.

Developed a double series with coefficients satisfying a q-summability condition.

Proved the series' universality concerning subseries and convergence modes.

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 $$\sum_{n,k=1}^\infty c_{n,k}\psi_n(x)\psi_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left | c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$ 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.