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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Bibliographic Details
Published in:Mathematische Nachrichten Vol. 291; no. 14-15; pp. 2204 - 2228
Main Authors: Ho, Vivian M., Ólafsson, Gestur
Format: Journal Article
Language:English
Published: Weinheim Wiley Subscription Services, Inc 01.10.2018
Subjects:
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
Online Access:Get full text
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Summary: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.
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content type line 14
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201600148