Martingale sequences in the theory of orthogonal series (Q1596186)

The sequence \(\{X_n,\mathcal F_n\), \(n=1,2,\dots\}\) of the random values \(X_n\) measurable with respect to the corresponding \(\sigma\)-algebras \(\mathcal F_n\) is called martingale, if for any \(m\) and \(n\), \( m\geq n\), \( X_n = E(X_m,\mathcal F_n)\). Besides, it is called closed on the right by the last element \(X_\infty\), if \( X_n = E(X_\infty,\mathcal F_n) \) for any~\(n\). The main result is as follows. A series in the Haar, Wolsh or multiplicative system is the Fourier series iff the corresponding martingale sequence of the partial sums (or its subsequence) is closed on the right by the finite element~\(f\).