On the existence of universal series by the generalized Walsh system

Abstract

In this paper, we prove the following: let $\omega \left(t\right)$ be a continuous function with $\omega (+0)=0$ and increasing in $[0,\infty )$. Then there exists a series of the form

$${\sum }_{k=1}^{\infty }{c}_k{\psi }_k\left(x\right)\mathrm{with}{\sum }_{k=1}^{\infty }{c}_k^2\omega \left(\right|c_k\left|\right)<\infty$$with the following property: for each $\epsilon >0$ a weight function $\mu \left(x\right)$, $0<\mu \left(x\right)\le 1$, $\left|\right\{x\in [0,1):\mu \left(x\right)\ne 1\left\}\right|<\epsilon$ can be constructed so that the series is universal in the weighted space $L_{\mu }^1[0,1)$ both with respect to rearrangements and subseries.