Prescribed Derivatives of Holomorphic Functions

ISSN： 0278-1077

Source： Complex Variables, Vol.48, Iss.2, 2003-01, pp. : 165-173

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Abstract

Let z₀ 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 alpha of order epsi Lambda (satisfying the obvious bounds implied by the radius of convergence for the maximal disc around z₀ in G) an analytic function f on G with $ f{( u )} (z_0) / u ! = alpha _ u $ for all epsi Lambda In that case, Lambda is called G-interpolating (in z₀). 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 z₀ in GThis enables us to characterize G-interpolating sets Lambda by a condition on the density of Lambda if, for instance, the intersection of G with the maximal circle around z₀ is an arc.