On interpolation problem by sums of exponentials series

Abstract

We formulate and study interpolation problems with infinite discrete sets of nodes in a convex domain by exponentials series which exponents are in a given unbounded set as well as by elements of subspaces that are invariant under differentiation in the space of holomorphic functions in a given convex domain or in all convex domains and with arbitrary interpolation nodes sets that are discrete in the domains. In the proofs, we use the well-known effect of forced analytic continuation of function used for interpolation. A condition on the set of exponents is obtained that is necessary and sufficient for resolvability of interpolation problem by elements of invariant subspaces that are generated by the exponents in all convex domains and with arbitrary interpolation nodes sets that are discrete in the domains. От the basis of this test it is proved that the interpolation problem may bу reduced to some equivalent problems e.g. to the problem of approximation of interpolation data by values of sums of exponentials series in the interpolation nodes. The existence of sums of exponentials series and functions in invariant subspaces that have complicated behavior near the boundary is proved