Unconditional convergence of general Fourier series

Abstract

S. Banach, in particular, proved that for any function, even $f(x) = 1,$ where $x\in[0,1],$ the convergence of its Fourier series with respect to the general orthonormal systems (ONS) is not guaranteed. In this paper, we find conditions for the functions $\varphi_{n}$ of an ONS $(\varphi_n),$ under which the Fourier series of functions $f\in Lip_1$ are unconditionally convergent almost everywhere. During our research, we mainly used the properties of the sequences of linear functionals on the Banach spaces to prove the main theorems we presented in this article. Our research has concluded that the aforementioned conditions do exist and are the best possible in a certain sense. We have also found that any ONS contains a subsystem such that the Fourier series of any function $f\in Lip_1$ is unconditionally convergent. Further, the precondition presented in Theorem \ref{theorem1.1}, which demands that the Fourier series of the function $f=1$ be convergent, is adequate, meaning that it does not make the actual conditions of Theorem \ref{theorem1.1} redundant. We have shown that the solution for these types of problems for the general ONS is trivial for the classical ONS (trigonometric, Haar, and Walsh systems).