Inequalities for Orthogonal Series and a Strengthening of the Carleman–Olevskii Theorem for Complete Orthonormal Systems

On the basis of interpolation theory, several new inequalities are established for both general orthonormal systems and various specific classes of orthonormal systems including the Haar and Franklin systems and wavelets. The solution of the problem of characterizing the Fourier coefficients of continuous functions for general orthonormal systems is completed. For every complete orthonormal system, a continuous function is constructed that generates a universal singularity similar to the one appearing in Carleman's theorem. This result significantly strengthens Olevskii's theorem and turns into Orlicz's theorem at the other end of the power scale. It is proved that the results obtained are, in a sense, final.