Differentiable functions and general orthonormal systems

The properties of the Fourier series of the functions from some differentiable class are well known for classical orthonormal systems (trigonometric, Haar, Walsh, etc.). On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). Therefore, in order to obtain well-known results for general ONS, we need to impose specific conditions on the given system. In the present paper we find conditions on the functions of an ONS under which the Fourier series of differentiable functions are convergent a.e.