598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors 598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 22E30, 43A85, 44A12, 45Q05, 92C55; Secondary 22E46, 32L25, 35S30, 65R32. Library of Congress Cataloging-in-Publication Data AMS Special Session on Radon Transforms and Geometric Analysis (2012 : Boston, Mass.) Geometric analysis and integral geometry : AMS special session in honor of Sigurdur Helgason’s 85th birthday, radon transforms and geometric analysis, January 4-7, 2012, Boston, MA ; Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces, January 8-9, 2012, Medford, MA / Eric Todd Quinto, Fulton Gonzalez, Jens Gerlach Christensen, editors. pages cm. – (Contemporary mathematics ; volume 598) Includes bibliographical references. ISBN 978-0-8218-8738-7 (alk. paper) 1. Radon transforms–Congresses. 2. Integral geometry–Congresses. 3. Geometric analysis– Congresses. I. Quinto, Eric Todd, 1951- editor of compilation. II. Gonzalez, Fulton, 1956- editor of compilation. III. Christensen, Jens Gerlach, 1975- editor of compilation. IV. Tufts University. Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces (2012 : Medford, Mass.) V. Title. QA672.A4726 2012 515.1–dc23 2013013624 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/598 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. 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Visit the AMS home page at http://www.ams.org/ 10987654321 131211100908 This volume is dedicated to Sigurdur Helgason whose mathematics has inspired many. v Contents Preface ix List of Presenters xi Historical Articles Some Personal Remarks on the Radon Transform Sigurdur Helgason 3 On the Life and Work of S. Helgason G. Olafsson´ and R. J. Stanton 21 Research and Expository Articles Microlocal Analysis of an Ultrasound Transform with Circular Source and Receiver Trajectories G. Ambartsoumian, J. Boman, V. P. Krishnan, and E. T. Quinto 45 Cuspidal discrete series for projective hyperbolic spaces Nils Byrial Andersen and Mogens Flensted-Jensen 59 The Radon transform on SO(3): motivations, generalizations, discretization Swanhild Bernstein and Isaac Z. Pesenson 77 Atomic decompositions of Besov spaces related to symmetric cones Jens Gerlach Christensen 97 A double fibration transform for complex projective space Michael Eastwood 111 Magnetic Schr¨odinger equation on compact symmetric spaces and the geodesic Radon transform of one forms Tomoyuki Kakehi 129 F -method for constructing equivariant differential operators Toshiyuki Kobayashi 139 Schiffer’s Conjecture, Interior Transmission Eigenvalues and Invisibility Cloaking: Singular Problem vs. Nonsingular Problem Hongyu Liu 147 vii viii CONTENTS Approximate Reconstruction from Circular and Spherical Mean Radon Transform Data W. R. Madych 155 Analytic and Group-Theoretic Aspects of the Cosine Transform G. Olafsson,´ A. Pasquale, and B. Rubin 167 Quantization of linear algebra and its application to integral geometry Hiroshi Oda and Toshio Oshima 189 Mean value theorems on symmetric spaces Franc¸ois Rouviere` 209 Semyanistyi fractional integrals and Radon transforms B. Rubin 221 Radon-Penrose transform between symmetric spaces Hideko Sekiguchi 239 Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations Joseph A. Wolf 257 Preface Geometric analysis on Euclidean and homogeneous spaces encompasses parts of representation theory, integral geometry, harmonic analysis, microlocal analysis, and partial differential equations. It is used in a wide array of applications in fields as diverse as inverse problems, tomography, and signal and data analysis. This volume provides articles giving historical perspectives, overviews of current research in these interrelated areas, and new results. We hope this motivates beginning researchers in these fields, and we wish that readers will be left with a good sense of important past work as well as current research in these exciting and active fields of mathematics. One theme of the volume is the geometric analysis motivated by the work of Sigurdur Helgason. An historical perspective is provided in the first article by Prof. Helgason himself and in the second article by Profs. Olafsson´ and Stanton. This emphasis is natural, since the volume is based, in part, on the AMS Special Session on Radon Transforms and Geometric Analysis in honor of Sigurdur Helgason’s 85th birthday held in Boston during the 2012 AMS Annual Meeting. Research papers related to this viewpoint, in particular, on Radon transforms and related mathematics are presented by Bernstein & Pesenson, Kakehi, Madych, Olafsson,´ Pasquale & Rubin, Rouvi`ere, Rubin, and others. The workshop on Geometric analysis on Euclidean and homogeneous spaces, held at Tufts University immediately following the AMS annual meeting, sought to expand on the topics presented at the special session. It was broader in scope, as evidenced by the contributions to this volume. Among contributions in pure mathematics are articles on representation the- ory and equivariant differential operators (Kobayashi and Oda & Oshima), Pen- rose transforms (Eastwood and Sekiguchi), wavelets related to symmetric cones (Christensen), representation theory and inductive limits of Lie groups (Wolf), and noncommutative harmonic analysis (Andersen & Flensted-Jensen). The interplay between integral geometry and applications is explored in the more applied articles. These include developing an elliptical Radon transform for ultrasound (Ambartsoumian, Boman, Krishnan & Quinto), using Schiffer’s conjec- ture to understand partial cloaking (Liu), and Radon transforms in crystallography (Bernstein & Pesenson) and thermoacoustic tomography (Madych). We thank the U.S. National Science Foundation and the Tufts University Dean’s Discretionary Fund for their support of the Tufts workshop. We are grateful to Tufts University Staff Assistant Megan Monaghan for the work she did behind the scenes to make the workshop successful. ix xPREFACE We thank the American Mathematical Society for its support of the Special Session honoring Sigurdur Helgason, and finally, we are indebted to Christine M. Thivierge, Associate Editor for Proceedings, for her indispensable role in making these proceedings a success. List of Presenters Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday American Mathematical Society National Meeting January 4-7 2012, Boston, MA Speakers Mark Agranovsky (Bar-Ilan University): Abel-Radon transform and CR functions. Nils Byrial Andersen, (Aarhus University, Denmark): Cusp Forms on hyperbolic spaces. Jan Boman (Stockholm University): Local injectivity of weighted Radon transforms. Jens Gerlach Christensen (Tufts University): Decomposition of spaces of distributions using G˚arding vectors. Susanna Dann (University of Missouri): Paley-Wiener Theorems on