Expansions in series of suitable systems of functions (Q1411552)

Let \(\Omega_R = \{z: |z|< R\}\), \(R > 0,\) and \(H (\Omega_R)\) be a set of all analytic functions in \(\Omega_R.\) Let \((f_k)_{k=0}^{\infty} \subset H(\Omega_R)\) be a system of such functions that each \(f_k\) can be represented in the form \[ f_k (z) = z^k \sum _{n=o}^{\infty}a_n^{(k)} z^n, z \in \Omega_R, \] and \(a_0^{(k)} = 1.\) In this paper it is considered the following problem. To find conditions, under which, for a suitable \(0 < r_1 < R\) each function \(f \in H(\Omega_{r_1})\) has a unique expansion of the form \[ f (z) = \sum _{k=0}^{\infty} c_k f_k (z), z \in \Omega_{r_0}, \] where \(0 < r_0 \leq r_1\) and the series converges absolutely uniformly on a compact set of \(\Omega_{r_0}.\) In Section 1 the author gives sufficient conditions for the solvability of this problem. In Section 2 he applies these results to prove expansions in terms of \(m\)-fold products of Bessel functions.