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

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 E<sup>2</sup>(ℂ\D)...

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Main Author: Delaney, William Kenneth
Format: Others
Language:en
Published: 1975
Online Access:https://thesis.library.caltech.edu/14348/1/Delaney_WK_1975.pdf
Delaney, William Kenneth (1975) A Counterexample in the Theory of Fourier Transforms in the Complex Domain. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8s0b-ck03. https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257 <https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257>
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spelling ndltd-CALTECH-oai-thesis.library.caltech.edu-143482021-09-01T05:05:44Z https://thesis.library.caltech.edu/14348/ A Counterexample in the Theory of Fourier Transforms in the Complex Domain Delaney, William Kenneth 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 E<sup>2</sup>(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. 1975 Thesis NonPeerReviewed application/pdf en other https://thesis.library.caltech.edu/14348/1/Delaney_WK_1975.pdf Delaney, William Kenneth (1975) A Counterexample in the Theory of Fourier Transforms in the Complex Domain. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8s0b-ck03. https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257 <https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257> https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257 CaltechTHESIS:08312021-161900257 10.7907/8s0b-ck03
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language en
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description 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 E<sup>2</sup>(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.
author Delaney, William Kenneth
spellingShingle Delaney, William Kenneth
A Counterexample in the Theory of Fourier Transforms in the Complex Domain
author_facet Delaney, William Kenneth
author_sort Delaney, William Kenneth
title A Counterexample in the Theory of Fourier Transforms in the Complex Domain
title_short A Counterexample in the Theory of Fourier Transforms in the Complex Domain
title_full A Counterexample in the Theory of Fourier Transforms in the Complex Domain
title_fullStr A Counterexample in the Theory of Fourier Transforms in the Complex Domain
title_full_unstemmed A Counterexample in the Theory of Fourier Transforms in the Complex Domain
title_sort counterexample in the theory of fourier transforms in the complex domain
publishDate 1975
url https://thesis.library.caltech.edu/14348/1/Delaney_WK_1975.pdf
Delaney, William Kenneth (1975) A Counterexample in the Theory of Fourier Transforms in the Complex Domain. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8s0b-ck03. https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257 <https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257>
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