A Counterexample in the Theory of Fourier Transforms in the Complex Domain
William Kenneth Delaney · CaltechTHESIS (California Institute of Technology) · 2021
The Borel transform of an entire function of exponential type is defined outside a closed bounded convex set D. Paley and Wiener have given a necessary and sufficient condition on the entire function F(z) such that φ(w), the Borel transform of F(z), is contained in E2(ℂ\D) for the case when D is a line segment. Kacnel'son has shown that the natural extension of this result provides a necessary condition for a general closed bounded convex set D. Here, by counterexample, we show that the natural extension does not provide a sufficient condition.