The Radon Transform
The first edition of this book has been out of print for some time and I have decided to follow the publisher's kind suggestion to prepare a new edition. Many examples with explicit inversion formulas and range theo rems have been added, and the group-th
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Sigurdur Helgason
The Radon Transform Second Edition
Progress in Mathematics Volume5
Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Sigurdur Helgason
The Radon Transform Second Edition
Springer Science+Business Media, LLC
Sigurdur Helgason Department of Mathematics MIT Cambridge, MA 02139
USA
Library of Congress Cataloging-in-Publication Data Helgason, Sigurdur, 1927 The Radon transform I Sigurdur Helgason. -- 2nd ed. p. em. -- (Progress in mathematics ; v. 5) Includes bibliographical references and index. ISBN 978-1-4757-1465-4 ISBN 978-1-4757-1463-0 (eBook) DOI 10.1007/978-1-4757-1463-0
1. Radon transforms. I. Title. II. Series: Progress in mathematics (Boston, Mass.) ; vol. 5. QA649.H44 1999 99-29331 515'. 723--dc21 CIP
AMS Subject Classifications: Primary: 22E30, 35L05, 43A85, 44A12, 53C65 Secondary: 22E46, 53C35, 92C55
Printed on acid-free paper. © 1999 Sigurdur Helgason, Second Edition Originally published by Birkhiiuser Boston in 1999
Birkhiiuser
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@1980 Sigurdur Helgason, First Edition
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ISBN 978-1-4757-1465-4 Typeset in ItXJEX by the author
98 765 43 21
SPIN 19901615
CONTENTS Preface to the Second Edition .................... ................. ix Preface to the First Edition .................... ................... xi
CHAPTER I The Radon Transform on JR.n
1. Introduction .................... .................... .............. 1 2. The Radon Transform of the Spaces V(JR.n) and S(JR.n). The Support Theorem .................... .................... .... 2 3. The Inversion Formula .................... .................... ... 15 4. The Plancherel Formula .................... .................... . 20 5. Radon Transform of Distributions .................... ........... 22 6. Integration over d-Planes. X-ray Transforms. The Range of the d-Plane Transform .................... ........ 28 7. Applications .................... .................... ............. 41 a) Partial Differential Equations .................... ........... 41 b) X-ray Reconstruction .................... ................... 46 Bibliographical Notes .................... .................... .... 51
CHAPTER II A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
1. Homogeneous Spaces in Duality .................... ............. 53 2. The Radon Transform for
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