Behavior of divergent Haar series (Q1068306)

The main result is the following: Let the series (*) \(\sum^{\infty}_{n=1} a_{i_ n}\chi_{i_ n}(x)\) be obtained as the result of an arbitrary permutation of the Haar series \(\sum^{\infty}_{n=1}a_ n\chi (x)\), satisfying \(a_ n\chi_ n(x)=o(n)\). Then on any interval \((\alpha,\beta)\subset [0,1]\) one can find a continuum of points at which (*) converges either to a finite value or to infinity of specific sign. This result is true, in particular, for Fourier-Haar series.