Smooth Functions and General Fourier Coefficients

As is known, the Fourier series of differentiable functions for classical orthonormal systems (trigonometric, Haar, Walsh, …) are absolutely convergent. However, for general orthonormal systems (ONSs) this fact does not hold. In the present paper, we consider some specific properties of special series of Fourier coefficients of differentiable functions with respect to the general ONSs. The results demonstrate that the properties of the general ONSs and of the subsequence of this system are essentially different. Here, we have shown that the received results are best possible.