On the recovery of the coefficients of a rearranged Haar series

The author has previously established that, for any rearrangement of a Haar series, if the rearranged series converges everywhere on $[0,1]$ to a bounded function $f(x)$ then the coefficients of the series can be recovered from the sum $f(x)$ using the formulas of Fourier; however, no analogous assertion holds, in general, for an arbitrary summable function $f$.
Generalizing his previous results, the author shows, in particular, that the theorem on the recovery of the coefficients of a rearranged Haar series goes over to functions of the class $L_2$, but not to the class $L_p$ for any $p<2$.
Bibliography: 9 titles.