Integral Geometry and Radon Transforms
In this text, integral geometry deals with Radon’s problem of representing a function on a manifold in terms of its integrals over certain submanifolds—hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental probl
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Sigurdur Helgason
Integral Geometry and Radon Transforms
1C
Sigurdur Helgason Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 USA [email protected]
ISBN 978-1-4419-6054-2 e-ISBN 978-1-4419-6055-9 DOI 10.1007/978-1-4419-6055-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938299 Mathematics Subject Classification (2010): 53C65, 44A12 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
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Preface
This book deals with a special subject in the wide field of Geometric Analysis. The subject has its origins in results by Funk [1913] and Radon [1917] determining, respectively, a symmetric function on the two-sphere S2 from its great circle integrals and an integrable function on R2 from its straight line integrals. (See References.) The first of these is related to a geometric theorem of Minkowski [1911] (see Ch. III, §1). While the above work of Funk and Radon lay dormant for a while, Fritz John revived the subject in important papers during the thirties and found significant applications to differential equations. More recent applications to X-ray technology and tomography have widened interest in the subject. This book originated with lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959–1965 on the Radon transform and its generalizations. The viewpoint of these generalizations is the following. The set of points on S2 and the set of great circles on S2 are both acted on transitively by the group O(3). Similarly, the set of points in R2 and the set P2 of lines in R2 are both homogeneous spaces of the group M(2) of rigid motions of R2 . This motivates our general Radon transform definition from [1965a] and [1966a], which forms the framework of Chapter II: Given two homogeneous spaces X = G/K and Ξ = G/H of the same group G, two elements x = gK and ξ = γH are said to be incident (denoted x#ξ) if gK ∩ γH = ∅ (as subsets of G). We then define the abstract Radon transform f → f from Cc (X) to C(Ξ) and the dual transform ϕ → ϕ from Cc (Ξ) to C(X) by ϕ(x) = f(ξ) = f (x) dm(x) , ϕ(ξ) dμ(ξ)
ˇ
ˇ
x#ξ
ξ#x
with canonical measures dm and dμ. These geometrically dual operators f → f and ϕ → ϕ are also adjoin
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