ON THE ALMOST EVERYWHERE CONVERGENCE OF NEGATIVE ORDER CESARO MEANS OF FOURIER–WALSH SERIES

Abstract

In the paper is presented existence of an increasing sequence of natural numbers $M_{\nu}, \nu=0,1,... ,$ such that for any $\varepsilon>0$ there exists a measurable set $E$ with a measure $\mu E>1-\varepsilon,$ such that for any function $f\in L^{1}[0,1]$ one can find a function $g\in L^{1}[0,1],$ which coincides with the function $f$ on $E$, and for any $\alpha \neq-1,-2,...$ the Cesaro means $\sigma_{M_{\nu}}^{\alpha}(x,\tilde{f}),~\nu=0,1,...,$ converges to $g(x)$ almost everywhere on [0,1].