Representation of Functions by Walsh's Series with Monotone Coefficients

Abstract

There exists a series in the Walsh system $\{\varphi_{n}\}$ of the form
\[
\sum_{i=1}^{\infty}a_{i}\varphi_{i},\quad\hbox{ with}\quad|a_{i}|\searrow0,
\]
that possess the following properties:

For any $\epsilon>0$ and any function $\displaystyle f\in L^{1}(0,1)$ there exists set $E\subset\lbrack0,1]$ $\left(  \left\vert E\right\vert >1-\epsilon\right)  $ and a sequence $\{\delta_{i}\}_{i=0}^{\infty},$ $\delta_{i}=0\ \hbox{or}\ 1$, such that the series
\[
\sum_{i=0}^{\infty}\delta_{i}a_{i}\varphi_{i}%
\]
converges to $f$ on $E$ in the $L^{1}(0,1)$-metric and on $[0,1]\diagdown E$ in the $L^{r}([0,1]\diagdown E)$ metric for all $r\in(0,1)$.