Paley–Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman–Opdam hypergeometric functions

Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundl...

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Veröffentlicht in:	Mathematische Nachrichten Jg. 291; H. 14-15; S. 2204 - 2228
Hauptverfasser:	Ho, Vivian M., Ólafsson, Gestur
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Weinheim Wiley Subscription Services, Inc 01.10.2018
Schlagworte:	43A85; Secondary: 22E46 43A90 53C35 Bundles Fourier transform Fourier transforms hypergeometric function Hypergeometric functions Integral transforms Paley–Wiener theorem Primary: 33C67 symmetric space Theorems
ISSN:	0025-584X, 1522-2616
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Abstract	Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space U/K. It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.
AbstractList	Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space U/K. It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix. Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K . An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.
Author	Ho, Vivian M. Ólafsson, Gestur
Author_xml	– sequence: 1 givenname: Vivian M. surname: Ho fullname: Ho, Vivian M. organization: Louisiana State University – sequence: 2 givenname: Gestur surname: Ólafsson fullname: Ólafsson, Gestur email: olafsson@math.lsu.edu organization: Louisiana State University
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SubjectTerms	43A85; Secondary: 22E46 43A90 53C35 Bundles Fourier transform Fourier transforms hypergeometric function Hypergeometric functions Integral transforms Paley–Wiener theorem Primary: 33C67 symmetric space Theorems
Title	Paley–Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman–Opdam hypergeometric functions
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