Convergence of Fourier series of functions $\text{Lip} 1$ with respect to general orthonormal systems

Abstract

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) $\{ \varphi_n(x)\}$ in order that the Fourier series in this system for any function from the class $\mathrm{L}\mathrm{i}\mathrm{p} 1$ be convergent almost everywhere on $[0, 1]$. It is shown that the obtained conditions are best possible in a certain sense.