%0 Journal Article
%T Representation of Functions by Walsh's Series with Monotone Coefficients
%A Razmik Melikbekyan
%J Journal of Mathematics Research
%D 2013
%I 
%R 10.5539/jmr.v5n1p107
%X There exists a series in the Walsh system ${varphi_{n}}$ of the form [ sum_{i=1}^{infty}a_{i}varphi_{i},quadhbox{ with}quad|a_{i}|searrow0, ] that possess the following properties: For any $epsilon>0$ and any function $displaystyle fin L^{1}(0,1)$ there exists set $Esubsetlbrack0,1]$ $left(  leftvert E ightvert >1-epsilon ight)  $ 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 $rin(0,1)$.
%U http://www.ccsenet.org/journal/index.php/jmr/article/view/24941