Unconditional convergence of general Fourier series

TL;DR

This paper establishes optimal conditions on orthonormal systems that guarantee the unconditional convergence of Fourier series for functions in Lip_1, extending classical results and identifying subsystems with desirable convergence properties.

Contribution

It introduces new conditions on orthonormal systems ensuring unconditional convergence of Fourier series for Lip_1 functions, and demonstrates their optimality and existence of suitable subsystems.

Findings

Conditions for unconditional convergence are proven to be optimal.

Any orthonormal system contains a subsystem with unconditional convergence.

Classical systems like trigonometric, Haar, and Walsh are trivial cases.

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).