Convergence of the Fourier Series of Lip 1 Functions with Respect to General Orthonormal Systems

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