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Expansions of Arbitrary Analytic Functions in Series of Exponentials

Published online by Cambridge University Press:  20 November 2018

D. G. Dickson
D. G. Dickson*
Affiliation:
University of Michigan, Ann Arbor, Michigan
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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 by

where 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 347 - 356
Copyright
Copyright © Canadian Mathematical Society 1981

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