Geodesic
Non-linear approximation of continuous functions
Sbornik. Mathematics, Tome 199 (2008) no. 5, pp. 629-653

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The existence of a function $f_0(x)\in C_{[0,1]}$ for which the greedy algorithm in the Faber-Schauder system is divergent in measure on $[0,1]$ is established. It is shown that for each $\varepsilon$, $0\varepsilon1$, there exists a measurable subset $E$ of $ [0,1]$ of measure $|E|>1-\varepsilon$ such that for each $f(x)\in C_{[0,1]}$ one can find a function $\widetilde f(x)\in C_{[0,1]}$ coinciding with $f(x)$ on $E$, whose greedy algorithm in the Faber-Schauder system converges uniformly on $[0,1]$. Bibliography: 33 titles. 
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     author = {M. G. Grigoryan and A. A. Sargsyan},
     title = {Non-linear approximation of continuous functions},
     journal = {Sbornik. Mathematics},
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     volume = {199},
     number = {5},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_5_a0/}
}
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M. G. Grigoryan; A. A. Sargsyan. Non-linear approximation of continuous functions. Sbornik. Mathematics, Tome 199 (2008) no. 5, pp. 629-653. http://geodesic.mathdoc.fr/item/SM_2008_199_5_a0/