Let D be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on D invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.
In the work we consider Dirichlet series. We study the problem of closedness for the set of the sums for such series in the space of functions holomorphic in a convex domain of a complex plane with a topology of uniform convergence on compact subsets. We obtain necessary and sufficient conditions under those each function in the closure of a linear span of exponents with positive indices is represented by a Dirichlet series. These conditions can be formulated only in terms of geometric characteristics of an index sequence and of the convex domain.
The existence of a basis is studied in a space of entire functions invariant under the differentiation operator. It is proved that every such space possesses a basis consisting of linear combinations of generalized eigenvectors. These linear combinations are formed within groups of exponents of arbitrarily small relative diameter. A complete description of the way to split the exponents into groups is obtained. Also, a criterion is found for the existence of a basis constructed by groups of zero relative diameter (so-called relatively small groups). In this connection a new criterion is obtained for the finiteness of the lower indicator of an entire function of exponential type.
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