Gap-Interpolation Theorems

Abstract

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(z_n) = w_n, 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 |z_n| → ∞ (and no z_n = 0).