A property of Fourier series

A theorem is proved making it possible, in certain cases, to use properties of the series $\sum_{k=1}^\infty c_k\varphi_k$ (where $\{\varphi_k\}$ is an orthonormal system in Hilbert space) to derive properties of the series $\sum_{k=1}^\infty f(c_k)\varphi_k$, where $f$ is a function of a complex variable, holomorphic in a region containing the origin and the points $c_1, c_2, \dots, c_k, \dots$, and such that $f (0)=0$.