An analog of the Menshov-Rademacher theorem for generalized orthosimilar systems (Q1596166)

The main result of the paper is the following assertion. Let \(X\) be a space with countable additive nonnegative measure, \(H=L^2(X)\), \(\{e^\omega(x)\}_{\omega\in\Omega}\) be a generalized orthosimilar nonnegative system of decompositions in \(H\), \(c(\omega)\) be a measurable function on \(\Omega\) with values in \({\mathbb R}\) or in \(\mathbb C\) (depending on the kind of field over which \(H\) is considered), and let \(\{\Omega_k\}_{k=1}^\infty\) be the exhaustion of \(\Omega\) such that the functions \(c(\omega) e_n^\omega\) are Lebesgue integrable on \(\Omega_k\), \(k, n\in\mathbb N\). If the function \(c(\omega)\) satisfies the condition \[ D = \Bigg(\int_{\Omega_1} + \sum_{k=2}^\infty \log_2^2 k\int_{\Omega_k\backslash\Omega_{k-1}}\Bigg) |c(\omega)|^2 d\mu(\omega)<\infty, \] then the sequence of the partial integrals \[ I_k(x) = \int_{\Omega_k} x(\omega)e^\omega(x) d\mu(\omega) \] converges almost everywhere on \(X\). Besides, the majorant \(I^*(x) = \sup_{k\in\mathbb N} |I_k(x)|\) of the sequence of partial integrals satisfies the estimate \[ \|I^*(x)\|\leq 3\sqrt{D}. \]