A generalization of a theorem of L. A. Balashov on subseries of the Fourier-Haar series.
For the sequence of integers \textbf{P }\( = \left\{ {p_i \geq 2} \right\}_1^\infty \) let us set \(m_0=1\), \(m_j = \prod_{s=1}^j {p_s}\) for \(j \in N\). Every number \(x \in \left[ {0,1} \right)\) can be written in the form \(x = \sum_{j = 1}^\infty {\frac{x_j }{m_j },}\) where \(x_j = 0,\dots,p_j - 1\) and for numbers with ambiguous (\textbf{P}-rational) representation we choose the sums which have a finite number of summands. The generalized Haar system \(\{\text{X}_n\}_0^\infty \) is defined on \([{0,1})\) as follows: \({\text{X}}_0 \left( x \right) \equiv 1\) and \[ {\text{X}}_k \left( x \right) = \left\{ \begin{matrix} \sqrt {m_n } \exp \left( {\frac{2\pi ix_{n + 1} }{p_{n + 1} }s} \right),\quad x \in \left[ {\frac{r}{m_n },\frac{r + 1}{m_n }} \right), \\ 0,\quad x \in \left[ {0,1} \right)\backslash \left[ {\frac{r}{m_n },\frac{r + 1}{m_n }} \right), \\ \end{matrix} \right. \] for \(k \geq 1\), where \(k = m_n + r( {p_{n + 1} - 1}) + s - 1,\;r = 0,\dots,m_n - 1,\;s = 1,\dots,p_{n + 1} - 1\). This system is orthonormal on \([{0,1})\). The main result of this article is the following Theorem 1. Let \(\sup_np_n < \infty\) and \(\{\text{X}_n\}_0^\infty\) be the generalized Haar system generated by the sequence \textbf{P}. Then for every Fourier series (1) \(\sum_{n=0}^\infty a_n\text{X}_n( x)\) with respect to this system and for every bounded sequence \(\{\varepsilon _n\}_0^\infty \) the series \(\sum_{n=0}^\infty \varepsilon_na_n \text{X}_n ( x)\) converges almost everywhere and is the Fourier series in the sense of the \(A\)-integral of its sum. In particular, any subsequence of the series (1) has this property. For the Haar system, that is for the case \(p_n \equiv 2\), this theorem was proved by L.A. Balashov (1987).
scientific article; zbMATH DE number 2177681