On Fourier coefficients of functions with respect to general orthonormal systems

We present results describing some properties of the Fourier coefficients of functions with respect to general orthonormal systems (ONS). We note that good differential properties of the functions do not ensure the ‘good’ behaviour of the Fourier coefficients (in the sense of convergence to zero) of these functions with respect to general ONS. We find conditions on the functions $\varphi_n(x)$ forming an ONS ($\varphi_n(x))$, $n=1,2,\dots$, for which the series of Fourier coefficients of the functions $f(x)$, where $f'(x)\in V(0,1)$, are absolutely convergent. We consider relationships between ONS, that is, problems of absolute independence for orthonormal systems.