| Abstract: | The paper proves that for any ε > 0 there exists ameasurable set E ? 0, 1] with measure |E| > 1 ? ε such that for each f ∈ L10, 1] there is a function \(\tilde f \in {L^1}\left {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L10, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \). |
