A Counterexample in the Theory of Fourier Transforms in the Complex Domain

Author: Delaney, William Kenneth

Year: 1975

Degree: Dissertation (Ph.D.)

Advisor: Luxemburg, W. A. J.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/8s0b-ck03

Abstract

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.

Files

• Delaney_WK_1975.pdf (application/pdf)