Nontrivial expansions of zero and representation of analytic functions by series of simple fractions

We propose a modification of the previously-known abstract scheme that reduces the problem of expansion of elements of a locally convex space in series over the system of eigenvectors of some linear operator to the question of existence of a nontrivial expansion of zero in this space. We implement this general scheme for the spaces of analytic functions in domains of the extended complex plane and the systems of simple fractions that are the eigenfunctions of the Pommier operator.