Non-linear approximation of continuous functions by the Faber-Schauder system

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 $\varepsilon$, $0<\varepsilon<1$, 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 $[0,1]$.
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