On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system

Abstract

‎Suppose that $\hat{b}_m\downarrow 0,\ \{\hat{b}_m\}_{m=1}^\infty\notin l^2,$‎ ‎and $b_n=2^{-\frac{m}{2}}\hat{b}_m$ for all $ n\in(2^m,2^{m+1}].$‎ ‎In this paper‎, ‎it is proved that any measurable and almost everywhere finite‎ ‎function $f(x)$ on $[0,1]$ can be corrected on a set of arbitrarily small measure‎ ‎to a bounded measurable function $\widetilde{f}(x)$; so that the nonzero Fourier-Haar‎ ‎coefficients of the corrected function present some subsequence of $\{b_n\}$‎, ‎and its‎ ‎Fourier-Haar series converges uniformly on $[0,1]$‎.