CoNTEMPORARY MATHEMATICS 63 Integral Geometry AMS-IMS-SIAM Summer Research Conference on Integral Geometry August 12-18, 1984 Bowdoin College, Brunswick, Maine Robert L. Bryant Sigurdur Helgason R. 0. Wells, Jr. Editors http://dx.doi.org/10.1090/conm/063 Recent Titles in This Series 184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995 183 William C. Couuett, Marc-Oiivier Gebubrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 199 5 182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jincbao Xu, Editors, Domain decomposition methods in scientific and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and quantization, 1994 178 Ht!Ume Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr., Editors, Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non"standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin, and Gregg J. Zuckerman, Editors, Mathematical aspects of conformal and topological field theories and quantum groups, 1994 174 Nancy Childress and John W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the spectrum, 1994 172 Peter E. Kloeden and Kenneth J. Palmer, Editors, Chaotic numerics, 1994 171 Riidiger GObel, Paul Hill, and Wolfgang Liebert, Editors, Abelian group theory and related topics, 1994 170 John K. Beem and Krishan L. Duggal, Editors, Differential geometry and mathematical physics, 1994 169 William Abikofr, Joan S. Birman, and Kathryn Kuiken, Editors, The mathematical legacy of Wilhelm Magnus, 1994 168 Gary L. Mullen and Peter Jau-Shyong Shiue, Editors, Finite fields: Theory, applications, and algorithms, 1994 167 RobertS. Doran, Editor, C*-algebras: 1943-1993, 1994 I 66 George E. Andrews, David M. Bressoud, and L. Alayne Parson, Editors, The Rademacher legacy to mathematics, 1994 165 Barry Mazur and Gleuu Stevens, Editors, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, I 994 164 Cameron Gordon, Yoav Moriah, and Bronislaw Wajnryb, Editors, Geometric topology, 1994 163 Zhong-Ci Shi and Chung-Chun Yang, Editors, Computational mathematics in China, 1994 162 Ciro Ciliberto, E. Laurs Livorni, and Andrew J. Sommese, Editors, Classification of algebraic varieties, 1994 161 Paul A. Schweitzer, S. J., Steven Hurder, Nathan Moreira dos Santos, and Jost! Luis Arrant, Editors, Differential topology, foliations, and group actions, I994 160 Niky Kamran and Peter J. Olver, Editors, Lie algebras, cohomology, and new applications to quantum mechanics, 1994 159 William J. Heinzer, Craig L. Huneke, and Judith D. Sally, Editors, Commutative algebra: Syzygies, multiplicities, and birational algebra, 1994 158 Eric M. Friedlander and Mark E. Mahowald, Editors, Topology and representation theory, 1994 (Continued in the back of this publication) CoNTEMPORARY MATHEMATICS 63 Integral Geometry AMS-IMS-SIAM Summer Research Conference on Integral Geometry August 12-18, 1984 Bowdoin College, Brunswick, Maine Robert L. Bryant Victor Guillemin Sigurdur Helgason R. 0. Wells, Jr. Editors American Mathematical Society Providence. Rhode Island Editorial Board Irwin Kra, W. H. Jaco managing editor Gerald J. Janusz Thomas F. Banchoff Jan Mycielski M. Salah Baouendi Johannes C. C. Nitsche Alan D. Weinstein The AMS-MAA-SIAM Joint Summer Research Conference in the Mathematical Sciences on Integral Geometry was held at Bowdoin College, Brunswick, Maine on August 12-18, 1984 with support from the National Science Foundation, Grant DMS-8218075. 1991 Mathematics Subject Classification. Primary 53C65; Secondary 53C35, 52A22, 60D05, 43A85. Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Integral Geometry {1984: Bowdoin College) Integral Geometry. (Contemporary mathematics, ISSN 0271-4132; v. 63) Bibliography: p. 1. Geometry, Integral-Congresses. I. Bryant, Robert L. II. American Mathematical Soci- ety. III. Institute of Mathematical Statistics. IV. Society for Industrial and Applied Mathe- matics. V. Title. VI. Series: Contemporary mathematics (American Mathematical Society); v. 63. QA649.A49 1984 516.3'62 86-28902 ISBN 0-8218-5071-7 (alk. paper) Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission«<math. ams. org. The appearance of the code on the first page of an article in this publication (including ab- stracts) indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. © Copyright 1987 by the American Mathematical Society. All rights reserved. All rights reserved except those granted to the United States Government. Printed in the United States of America. This volume was printed from author prepared copy. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 0 Printed on recycled paper. 10 9 8 7 6 54 3 2 00 99 98 97 96 95 TABLE OF CONTENTS Preface vii Spectral synthesis on symmetric spaces By Carlos A. Berenstein 1 The finite Radon transform By Ethan D. Balker 27 Hyperfunctions in representation theory and mathematical phys~cs By Edward G. Dunne 51 The exponential Radon transform By David V. Finch and Alexander Hertle 67 Integral geometry as geometry and as analysis By S. G. Gindikin 75 Euclidean Radon transforms: ranges and restrictions By Eric L. Grinberg 109 Perspectives in integral geometry By Victor Guillemin 135 Some results on Radon transforms, Huygen's principle and x-ray transforms By Sigurdur Helgason 151 Classical integral geometry in Riemannian homogeneous spaces By Ralph Howard 179 Differential operators and Cartan motion groups By Kenneth D. Johnson 205 v vi TABLE OF CONTENTS An L2-cohomology analogue of the Penrose transform for the oscillator representation By Lisa A. Mantini 221 Injectivity of rotation invariant Radon transforms on complex hyperplanes in en By Eric Todd Quinto 245 On overdetermined systems associated with integral geometry transforms in the real projective space By Radu RO§U 261 The Pompeiu problem in exterior domains in symmetric spaces By Mehrdad Shahshahani and Alladi Sitaram 267 Curvature integrals and Chern classes of singular varieties By Theodore Shifrin 279 Hypothesis testing in integral geometry: guessing the shape of a plane domain By Peter Waksman 299 Non-linear integral transforms By Richard S. Ward 307 Integral geometry and twistor theory By R. 0. Wells, Jr. 317 Some inverse problems of potential theory By Lawrence Zalcman 337 PREFACE In the summer of 1984 a one-week conference on Integral Geom- etry was held at Bowdoin College in Brunswick, Maine. It was one of a series of conferences organized under the auspices of the AMS-SIAM-IMS Summer Research Conference Series. The organizing committee consisted of Robert Bryant (chairman), Victor Guillemin, Sigurdur Helgason, and R. 0. Wells, Jr. The papers in this volume are contributions by participants whose papers, after refereeing, were considered suitable for inclusion. They range from purely expository to quite technical papers, and as a whole represent a good survey of contemporary work in this area. The topic of Integral Geometry as such is not nearly so uni- versally known as its counterpart, Differential Geometry, and most of the major results in this area are part of the twentieth cen- tury, while Differential Geometry has its very strong nineteenth century tradition going back to Gauss and Riemann. The conference brought together researchers from three differ- ent areas of contemporary research in topics involving integral geometry. The first area deals with new developments concerning· the classical problems of computing geometric invariants by statis- tical averaging procedures (to oversimplify greatly). The second area relates to a circle of ideas concerning what is now called the Radon transform, going back to the seminal work of Funk and Radon around 1916-1917. The third area deals with integral- viii PREFACE geometric transforms which are now being used in the study of field equations in mathematical physics, work which derives