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Lecture Notes in Mathematics 1684 Editors: A. Dold, Heidelberg F. Takens, Groningen Subseries: Fondazione C. I. M. E., Firenze Advisor: Roberto Conti

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Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

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C. A. Berenstein R E Ebenfelt S.G. Gindikin S. Helgason A.E. Tumanov Integral Geometry, Radon Transforms and Complex Analysis Lectures given at the 1s t Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venice, Italy, June 3-12, 1996 Editors: E. Casadio Tarabusi, M. A. Picardello, G. Zampieri Fondazione C.I.M.E. Springer

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Authors Editors Carlos A. Berenstein Enrico Casadio Tarabusi Institute for Systems Research Dipartimento di Matematica 221 A. V. Williams Building "G. Castelnuovo" University of Maryland Universith di Roma College Park, MD 20742-0001, USA "La Sapienza" Piazzale Aldo Moro, 2 Peter F. Ebenfelt 00185 Roma, Italy Department of Mathematics Royal Institute of Technology Massimo A. Picardello 100 44 Stockholm, Sweden Dipartimento di Matematica Universith di Roma "Tor Vergata" Simon Gindikin Via della Ricerca Scientifica Department of Mathematics 00133 Roma, Italy Hill Center Rutgers University Giuseppe Zampieri New Brunswick, NJ 08903-2101, USA Dipartimento di Matematica Pura ed Applicata Sigurdur Helgason Universit'~ di Padova Department of Mathematics Via Belzoni, 7 Massachusetts Institute of Technology 1-35131 Padova, Italy Cambridge, MA 02139-4307, USA Alexander Tumanov Department of Mathematics University of Illinois 1409 West Green Street Urbana-Champaign, IL 61801-2943, USA Cataloging-in-PublicationD ata applied for Die Deutsche Bibtiothek - CIP-Einheitsaufnahme Integral geometry, radon transforms and complex analysis : held in Venezia, Italy, June 3-12. 1996 / C. A. Berenstein ... Ed.: E. Casadio Tarabusi ... - Berlin; Heidelberg; New York; Barcelona: Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998 (Lectures given at the ...session of the Centro lnternazionale Matematico Estivo (CIME) ... ; 1996,1) (Lecture notes in mathematics; vol. 1684: Subseries; Fondazione CIME) ISBN 3-540-64207-2 Centro Internationale Matematico Estivo <Firenze>: Lectures given at the ... session of the Centro lnternationale Matematico Estivo (CIME) ... - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong: Springer Friiher Schriftenreihe. - FriJher angezeigt u. d. T.: Centro lnternationale Matematico Estivo: Proceedings of the ... session of the Centro l nternationale Matematico Estivo (C1ME) 1996,1. Integral geometry, radon transforms and complex analysis. - 1998 Mathematics Subject Classification (1991): 43-06, 44-06, 32-06 ISSN 0075- 8434 ISBN 3-540-64207-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. �9 Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10649783 46/3143-543210 - Printed on acid-free paper

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PREFACE This book contains the notes of five short courses delivered at the Italian Can- fro Internazionale Matematico Estivo (CIME) session Integral Geometry, Radon Transforms and Complex Analysis held at Ca' Dolfin in Venice (Italy) in June 1996. Three of the courses (namely those by: Carlos A. Berenstein of the University of Maryland at College Park; Sigurdur Helgason of the Massachusetts Institute of Technology; and Simon G. Gindikin of Rutgers University) dealt with various aspects of integral geometry, with a common emphasis on several kinds of Radon transforms, their properties and applications. The lectures by C. A. Berenste in , Radon transforms, wavelets, and applica- tions, explain the definition and properties of the classical Radon transform on the two-dimensional Euclidean space, with particular stress on localization and inver- sion, which can be achieved by the recent tool of wavelets. Interesting applications to Electrical Impedance Tomography (EIT) are also illustrated. The lectures by S. Helgason, Radon transforms and wave equations~ give an account of Radon transforms on Euclidean and symmetric spaces, focusing atten- tion onto the Huygens principle and the solution of the wave equation in these environments. The lectures by S. G. Gindikin, Real integral geometry and complex analysis, give an account of the deep connection between the two main themes of this CIME session, covering several variations of the Radon transform (RT): the projective RT; RT's taken over hyperplanes of codimension higher than 1; and RT's over spheres. An important and unifying tool is the ~" operator of Gel'fand-Graev-Shapiro, used to explain analogies between inversion formulas for the various RT's. This approach goes hand-in-hand with ~-cohomotogy and hyperfunctions, typical subjects in the field of complex a~alysis. In related areas, the other two courses (namely those by: Alexander E. Tumanov of the University of Illinois at Urbana-Champaign; Peter F. Ebenfelt of the Royal Institute of Technology at Stockholm) share stress on CR manifolds and related problems. The lectures by A. E. Tumanov , Analytic discs and the extendibility of CR functions, provide an introduction to CR structures and deal in particular with the problem of characterizing those submanifolds of C N whose CR functions are wedge-extendible. This property turns out to be equivalent to the absence of proper submanifolds which carry the stone CR structure. (The technique of the proof con- sists in an infinitesimal deformation of analytic discs attached to CR submanifolds.) The lectures by P. F. Ebenfel t , Holomorphic mappings between real analytic aubmanifolds in complex space, deal with algebralcity of locally invertible holomor- phic mappings. Along with classical results, new criteria are introduced in terms of the behavior of these mappings on a real-analytic CR submanifold which is generic, minimal, and holomorphically non-degenerate in a suitable sense. To this end a fundamental tool is afforded by the so-called Segre sets.

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VI We wish to express our appreciation to the authors of these notes, and to thank all the numerous participants of this CIME session for creating a lively and stim- ulating atmosphere. We are particularly grateful to those who contributed to the success of the session by delivering very inspiring talks. Enrico CASADIO TARABUSI Massimo A. P I C A R D E L L O Giuseppe ZAMPIERI

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TABLE OF CONTENTS BERENSTEIN, C. A. Radon Transforms, Wavelets, and Applications EBENFELT, P. F. Holomorphic Mappings Between Real Analytic Submanifolds in Complex Space 35 GINDIKIN, S. G. Real Integral Geometry and Complex Analysis 70 HELGASON, S. Radon Transforms and Wave Equations 99 TUMANOV, A. E. Analytic Discs and The Extendibility of CR Functions 123

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Radon transforms, wavelets, and applications Carlos Berenstein We present here the informal notes of four lectures 1 given at Cs Dolfin, Venice, under the auspices of CIME. They reflect the research of the author, his collaborators, and many other people in different applications of integral geometry. This is a vast and very active area of mathematics, and we try to show it has many diverse and sometimes unexpected applications, for that reason it would impossible to be complete in the references. Nevertheless, we hope that every work relevant to these lectures, however indirectly, will either be explicitly found in the bibliography at the end or at least in the reference lists of the referenced items. I apologize in advance for any shortcomings in this respect. The audience of the lectures was composed predominantly of graduate students of universities across Italy and elsewhere in Europe, for that reason, the emphasis is not so much in rigor but in creating an understanding of the subject, good enough to be aware of its manifold applications. There are several very good general references, the most accesible to students is, in my view, Hell. For deeper analysis of the Radon transform the reader is suggested to look in He2 and He3. For a very clear explanation of the numerical algorithms of the (codimension one) Radon transform in R 2 and R 3, see Na and KS. There have also been many recent conferences on the subject of these lectures, for a glimpse into them we suggest GG and GM. Finally, I would like to thank the organizers, Enrico Casadio Tarabusi, Massimo Picardello, and Giuseppe Zampieri, for their kindness in inviting me and for the effort they exerted on the organization of this CIME session. I am also grateful to David Walnut for suggestions that improved noticeably these notes. 1. T o m o g r a p h i c imag ing of space p l a s m a Space plasma is composed of electrically charged particles that are not uniformly distributed in space and are influenced by celestial bodies. The problem consists in determining the distribution function of the energy of these particles (or of their velocities) in a region of space. A typical measuring device will take discrete mea- surements (for instance, sample temperatures at different points in space) and then the astrophysicist will try to fit a "physically meaningful" function passing through these points. The procedure proposed in ZCMB is based on the idea that the mea- surements should directly determine the distribution function. We do it by exploiting the charged nature of the particles and using the Radon transform. (The recently launched Wind satellite carries a measuring device based on similar interaction prin- ciples and requires tomographic ideas for the processing of the data.) The advantage of the tomographic principles that we shall describe presently is that each measurement carries global information and seems to have certain noise reduction advantages over the pointwise measurements of temperatures, which is the 1These lectures reflect research of the author partially supported by the National Science Foundation.

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| ( Figure 1: Schematic detector. usual technology. We will describe everything in a two-dimensional setting, but the more realistic three-dimensional case can be handled similarly. The instrument we proposed in ZCMB is schematically the following. An electron enters into the instrument (a rectangular box in the figure below) through an opening located at the origin and is deflected by a constant magnetic field/~ perpendicular to the plane of the paper (see Figure 1). Under the Lorentz force, the electrons follow circular orbits and strike detectors lo- cated on the front-inside surface of the box (along the y axis). Those that strike a detector located at the point y have the property that where m is the mass of the electron, e its charge, and B the magnitude of the magnetic field /~. In other words, all the electrons with the same first component v~ of their velocities strike the same detector located at the height y. The range of velocities over a segment of width a (width of the detector) is Avx = (eB/2m)d (In terms of the length of the detector plate D in Figure 1 and the maximum velocity vm~ we have Avx = (d/D) - Vm~). If f(v~,vy) represents the electron velocity dis- tribution, then the number dN of electrons counted by a detector in time dt is given by dN = Anev~Avx i f(vx, vy)dvy, dr, - - 0 0 ne is electron density and A is the area of the entrance aperture. In other words, 1 dN f(v~, v~)dvy = AneAv~--~ - -o o

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so tha t the count of hits provides the integral of f along a line vx = constant in the velocity plane. By rota t ing the detector or changing the orientation of the magnet ic field we obtain the Radon transform of f . As a realistic example, consider a p lasma of nominal electron density ne = 10 c m -3 , velocity in the range Vmin to Vmax of 1.2 X 10 s to 3.0 • 109 cm s -1, average velocity = 6.5 • 108 cm s -1, and we assume a Gaussian distr ibution function so tha t dt - const, e x p , , 2~2, with individual detector area and aper ture of 0.04 c m 2 for a small instrument one gets tha t the dis tr ibut ion function f varies from 1 to 10 -5 while d N / d t varies from 102 to 105s -1. The s tandard measurement methods make the a priori assumption tha t f is the sum of a Gaussian centered at V and per turbed by adding a finite collection of Gaussians, often located in the region where f varies from 10 -4 to 10 -5, but the previously described instrument does not require any such assumption, on the other hand, experimental ly one sees that such large variations, like from 1 to 10 -5 as in the example, are realistic. We shall see in Section 2 tha t this is an embodiment of the Radon transform in R 2. The more realistic case of 3-d is handled by an instrument where there is a plane which contains the entrance aper ture and a 2-d array of detectors in the plane (x, y). One shows tha t at each detector location (x, y) one obtains an integral over a planar curve and that the addi t ion of overall elements with the same x component leads to a 2-d plane integral of the density dis t r ibut ion so tha t we have the Radon transform in R 3. (This is an observation we made joint ly with M. Shahshahani.) Before concluding this section, let us remark tha t the large variations expected from the velocity density function f make the inversion of the Radon transform very ill-conditioned, even if f is assumed to be a smooth function. This is due to the continuity propert ies of the Radon transform and its inverse as seen in the next section. The remarkable point is that in medical applications, like CAT scans, the unknown density is natural ly discontinuous along some curves but otherwise it has small local variations, and it is this reason the inversion problem is u l t imate ly easier for medical applications.

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