On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system (Q1790514)

The authors are concerned with the behaviour of Fourier series in the Haar system. It is proved that there are measurable sets $E$, of arbitrarily small measure, such that a suitable change of the values of any function of class $L^\infty[0, 1]$ on $E$ leads to a new modified function $g\in L^\infty[0, 1]$, whose Fourier series in the Haar system converges uniformly on $[0, 1]$ and the nonzero elements in the sequence constructed by the $L^\infty$ norms of the elements in its unique Fourier-Haar series are monotonically decreasing.