Modifications of functions, Fourier coefficients and nonlinear approximation

M. G. Grigoryan

Sbornik. Mathematics, Tome 203 (2012) no. 3, pp. 351-379

Résumé

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.

Détail

Keywords: correction of functions, nonlinear approximation, greedy algorithm.
Mots-clés : Fourier coefficients

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