Abstract:
This work continues the author's investigations of the convergence of greedy algorithms from the
standpoint of classical results on correction of functions. In particular, the following result is obtained: for each $\varepsilon$, $0<\varepsilon<1$, there exists a measurable set $E\subset [0,1)$ of measure
$|E|>1-\varepsilon$ such that for each function $f\in L^{1}[0,1)$ a function $\widetilde{f}\in L^{1}(0,1)$ equal to $f$ on $E$ can be found such that the greedy algorithm for $\widetilde{f}$ with respect to the Walsh system converges to it almost everywhere on $[0,1]$, and all the nonzero elements of the sequence of
Walsh-Fourier coefficients of the function thus obtained are arranged in decreasing order of their absolute values.
Bibliography: 35 titles.
Keywords:Fourier coefficients, correction of functions, nonlinear approximation, greedy algorithm.