Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions

We consider a series with respect to a multiplicative Price system or a generalized Haar system and assume that the martingale subsequence of its partial sums converges almost everywhere. In this paper we prove that, under certain conditions imposed on the majorant of this sequence, the series is a Fourier series in the sense of the $A$-integral (or its generalizations) of the limit function if the series is considered as a series with respect to a system with $\sup p_n<\infty$. In similar terms, we also present sufficient conditions for a series to be a Fourier series in the sense of the usual Lebesgue integral. We give an example showing that the corresponding assertions do not hold if $\sup p_n=\infty$.