Laplace transform representations and Paley-Wiener theorems for functions on vertical strips
type: Item
instance of: scholarly article
title: Laplace transform representations and Paley-Wiener theorems for functions on vertical strips
zbMATH Open document ID: 1203.30040
author: Zen Harper
published in: Documenta Mathematica
publication date: 2010-04-30T00:00:00Z
full work available at URL: https://eudml.org/doc/231189
full work available at URL: 06.html
review text: The author works on a very classical problem, but nevertheless succeeds in obtaining very meaningful results. Let \(F(z)\) be a holomorphic function on a strip \(\{a<\text{Re}(z)< b\}\). The problem is to determine when this function can be represented as a bilateral Laplace transform \[ F(z)= \int^\infty_{-\infty} e^{-zt} f(t)\,dt \] in some reasonable sense. The main result is a Paley-Wiener type theorem which gives sufficient conditions for the existence of \(f(t)\) for functions \(F\) belonging to a certain weighted Bergman space. The author considers the representation problem also in the case of bounded functions on half-planes and extends a result of \textit{Ch. K. Batty} and \textit{M. D. Blake} [C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 2, 71--75 (2000; Zbl 0945.44001)]. The paper ends with a discussion of the uniqueness of these representations. If the function \(F(z)\) is represented by \(f_1\) and \(f_2\) on two disjoint strips, but extends analytically between them, when is \(f_1= f_2\)? The author shows that this question is related to the uniqueness of the Cauchy problem for the heat equation with complex space variable.
reviewed by: Khristo N. Boyadzhiev
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30H20
zbMATH DE Number: 5703146
zbMATH Keywords: Laplace transform
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: heat equation
zbMATH Keywords: maximum principle
zbMATH Keywords: analytic continuation
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Bergman spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5703146