Abstract
A theorem is proved making it possible, in certain cases, to use properties of the series $$\sum\nolimits_k^\infty { = _1 c_k \varphi _k } $$ (whereϕ k is an orthonormal system in Hilbert space) to derive properties of the series $$\sum\nolimits_k^\infty { = _1 f(c_k )\varphi } _k $$ , wheref is a function of a complex variable, holomorphic in a region containing the origin and the points c1, c2,..., ck,..., and such thatf (0)=0.
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