Some problems of convergence of general Fourier series

TL;DR

This paper investigates conditions under which modified Fourier series with weighted coefficients converge almost everywhere, demonstrating the optimality of these conditions and exploring the limits of convergence for general orthonormal systems.

Contribution

It establishes the best possible conditions on coefficient sequences ensuring almost everywhere convergence of Fourier series with bounded variation functions.

Findings

Identifies optimal conditions on sequences for convergence

Shows these conditions are the best possible

Extends understanding of Fourier series convergence limits

Abstract

S. Banach \cite{Banach} proved that good differential properties of function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). On the other hand it is very well known that a sufficient condition for the a.e. convergence of an orthonormal series is given by the Menshov-Rademacher Theorem. The paper deals with sequence of positive numbers $(d_n)$ such that multiplying the Fourier coefficients $(C_n(f))$ of functions with bounded variation by these numbers one obtains a.e. convergent series of the form $\sum_{n=1}^{\infty}d_n C_n(f) \varphi_n (x).$ It is established that the resulting conditions are best possible.