Prescribed Derivatives of Holomorphic Functions

Abstract

Let z 0 be a point in an open set $ G \\subseteq {\\shadC} $ and $ \\Lambda \\subseteq {\\shadN}_0 $ an infinite set. We study the problem when it is possible to find for all prescribed derivatives f x of order x k v (satisfying the obvious bounds implied by the radius of convergence for the maximal disc around z 0 in G ) an analytic function f on G with $ f^{(\ u )} (z_0) / \ u ! = \\alpha _\ u $ for all x k v In that case, v is called G -interpolating (in z 0 ). We prove by functional analytic methods (a variation of the Banach-Schauder open mapping theorem and Köthe's description of the dual of the Fréchet space H(G) ) that this property only depends on the intersection of G with the maximal circle around z 0 in G This enables us to characterize G -interpolating sets v by a condition on the density of v if, for instance, the intersection of G with the maximal circle around z 0 is an arc.