Expansions of Arbitrary Analytic Functions in Series of Exponentials

Abstract

Let ϕ ≠ 0 be an entire function of one complex variable and of exponential type. Let B denote the set of all monomial exponentials of the form zneζ where ζ is a zero of ϕ of order greater than h. If R is a simply connected plane region and H(R) denotes the space of functions analytic in R with the topology of uniform convergence on compacta, then ϕ can be considered as an element of the topological dual H′(R) if the Borel transform of ϕ is analytic on , the complement of R. The duality is given bywhere C is a simple closed curve in the common region of analyticity of ƒ and , and C winds once around the complement of a set in which is analytic.