Quasiunconditional basis property of the Faber – Schauder system

Abstract

We prove that, for any $0 < \delta < 1$, there exists a measurable set $E_{\delta} \subset [0, 1], \mathrm{m}\mathrm{e}\mathrm{s} (E_{\delta }) > 1 \delta $, such that for any function $f \in C[0, 1]$, one can find a function $\widetilde f \in C[0, 1]$ that coincides with f on E\delta , and the Fourier – Faber – Schauder series for the function $\widetilde f$ unconditionally converges in $C[0, 1]$. Moreover, the moduli of the nonzero Fourier – Faber – Schauder coefficients of the function $\widetilde f$ coincide with the elements of a given sequence $\{ b_n\}$ satisfying the condition $$b_n \downarrow 0,\; \sum^{\infty }_{n=1} frac{b_n}{n} = +\infty .$$