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Transactions of the American Mathematical Society

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Real Paley-Wiener theorems and local spectral radius formulas

Authors: Nils Byrial Andersen and Marcel de Jeu
Journal: Trans. Amer. Math. Soc. 362 (2010), 3613-3640
MSC (2010): Primary 42B10; Secondary 47A11
Published electronically: February 15, 2010
MathSciNet review: 2601602
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Abstract: We systematically develop real Paley-Wiener theory for the Fourier transform on $ \mathbb{R}^d$ for Schwartz functions, $ L^p$-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley-Wiener theory is included.

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Additional Information

Nils Byrial Andersen
Affiliation: Mads Clausen Institute, University of Southern Denmark, Alsion 2, DK-6400 Sønderborg, Denmark
Address at time of publication: Alssundgymnasiet Sønderborg, Grundtvigs Allé 86, 6400 Sønderborg, Denmark

Marcel de Jeu
Affiliation: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

Keywords: Paley--Wiener theorem, Fourier transform, constant coefficient differential operator, local spectrum, local spectral radius formula
Received by editor(s): May 12, 2008
Published electronically: February 15, 2010
Additional Notes: The first author was supported by a research grant from the European Commission IHP Network: 2002–2006 Harmonic Analysis and Related Problems (Contract Number: HPRN-CT-2001-00273 - HARP)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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