Convergence of General Fourier Series of Differentiable Functions

Abstract

Convergence of classical Fourier series (trigonometric, Haar, Walsh, \(\dots\) systems) of differentiable functions are trivial problems and they are well known. But general Fourier series, as it is known, even for the function \(f(x)=1\) does not converge. In such a case, if we want differentiable functions with respect to the general orthonormal system (ONS) \((\varphi_{n})\) to have convergent Fourier series, we must find the special conditions on the functions \(\varphi_{n}\) of system \((\varphi_{n})\). This problem is studied in the present paper. It is established that the resulting conditions are best possible. Subsystems of general orthonormal systems are considered.