1984 | OriginalPaper | Chapter
Gap-Interpolation Theorems
Authors : D. H. Luecking, L. A. Rubel
Published in: Complex Analysis
Publisher: Springer New York
Included in: Professional Book Archive
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There are many theorems in classical analysis where gaps play a rôle. We take up now some considerations from [N. Kalton and L. A. Rubel] where gaps and interpolation are mixed. The idea is to take the Germay interpolation situation, where we want f(zn) = wn, n = 1,2,3,… for some entire function f but now require that f have the form $$ f(z) = \mathop{\Sigma }\limits_{{\lambda \in \Lambda }} \,{a_{\lambda }}{z^{\lambda }} $$ where ⋀ is a given set of positive integers. For certain ⋀ (like ⋀ = $$ \Lambda = \mathbb{N} $$, the set of all positive integers), this interpolation is always possible—provided we require |zn| → ∞ (and no zn = 0).