Interpolation and fundamental principle

Abstract

In this work we study the spaces of functions analytic in convex domains in the complex plane. We consider subspaces of such spaces, which are invariant with respect to the differentiation operator. We study the fundamental principle problem for an invariant subspace, that is, the problem on representing all its elements by a series of eigenfunctions and generalized eigenfunctions of the differentiation operator in this subspace, which are the exponentials and exponential monomials. We provide a complete description of the space of sequences of the coefficients of the series, by which we represent the functions from the invariant subspace. We also study the multiple interpolation problem in the spaces of entire functions of exponential type. We consider the duality of interpolation problem and fundamental principle. This duality problem is completely solved. We established the duality of the fundamental principle problem and interpolation problem for an arbitrary convex domain with no restrictions.