General Fourier coefficients and convergence almost everywhere

We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system $(\varphi_n)$ in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men'shov–Rademacher theorem. We also prove a theorem saying that every system $(\varphi_n)$ contains a subsystem $(\varphi_{n_k})$ with respect to which the Fourier coefficients of functions of bounded variation satisfy those hypotheses. The results obtained complement and generalize the corresponding results in [1].