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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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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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> |
work_keys_str_mv |
AT delaneywilliamkenneth acounterexampleinthetheoryoffouriertransformsinthecomplexdomain AT delaneywilliamkenneth counterexampleinthetheoryoffouriertransformsinthecomplexdomain |
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