Table Of ContentHarmonic analysis
and integral
geometry
CHAPMAN & HALL/CRC
Research Notes in Mathematics Series
Main Editors
H. Brezis, Universite de Paris
R.G. Douglas, Texas A&M University
A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)
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H. Amann, University of Zurich B. Moodie, University of Alberta
R. Aris, University of Minnesota S. Mori, Kyoto University
G. I. Barenblatt, University of Cambridge L.E. Payne, Cornell University
H. Begehr, Freie Universitat Berlin D.B. Pearson, University of Hull
R Bullen, University of British Columbia 1. Raeburn, University of Newcastle, Australia
R.J. Elliott, University of Alberta G.F. Roach, University of Strathclyde
R.P. Gilbert, University of Delaware I. Stakgold, University of Delaware
D. Jerison, Massachusetts Institute of Technology W.A. Strauss, Brown University
B. Lawson, State University of New York J. van der Hoek, University of Adelaide
at Stony Brook
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Massimo A Picardello
Harmonic analysis
and integral
geometry
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Library of Congress Cataloging-in-Publication Data
Harmonic analysis and integral geometry I edited by Massimo A. Picardello.
p. cm.--(Chapman & HalUCRC research notes in mathematics series; 422)
Includes bibliographical references.
ISBN 1-58488-183-6 (alk. paper)
I. Harmonic analysis--Congresses. 2. Integral geometry--Congresses. I. Picardello,
Massimo A.. 1949-II. Series.
QA403 .H2225 2000
515' .2433--dc21 00-055484
Library of Congress Card Number 00-055484
Preface
This volume contains the proceedings of the first Summer University of
Safi, Morocco. The Summer University of Safi is an annual advanced re
search school and congress, supported by several institutions in Morocco,
with a significant contribution by the University “Hassan II” of Casablanca.
The beginning sessions were devoted to mathematical research. The sub
ject of the first session, held July 15-20, 1998, was “Harmonic Analysis and
Integral Geometry.” It was a lively and very successful scientific event, at
tended by many of the research leaders in this field, who gave individual
lectures and courses. The extensive participation of mathematicians from
Maghreb, whose conferences gave full evidence of the advanced level of local
research in mathematics, confirms the continuing tradition of mathematical
excellence in North Africa that began thousands of years ago.
Besides the contributors to this volume, many other participants held
conferences at the school. To them, and to all the other participants, I wish
to express my gratitude for the stimulating scientific environment that they
helped create.
Massimo Picardello
Contents
Preface
Fulton B. Gonzalez
John’s Equation and the Plane to Line Transform on R3..............1
Tomoyuki Kakehi
Radon Transforms on Compact Grassmann Manifolds
and Invariant Differential Operators of Determinantal Type.......9
Takaaki Nomura
Invariant Berezin Transforms................................................................19
Simon Gindikin
Integral Geometry on Hyperbolic Spaces.........................................41
Massimo A. Picardello
The Geodesic Radon Transform on Trees.........................................47
Enrico Casadio Tarabusi, Joel M. Cohen and Flavia Colonna
The Distribution-Valued Horocyclic Radon Transform
on Trees.........................................................................................................55
Laura Atanasi
Integral Geometry on Affine Buildings...............................................67
Ahmed Abouelaz
Integral Geometry in the Sphere Sd....................................................83
Andrea D’Agnolo and Corrado Marastoni
A Topological Obstruction for the Real Radon Transform.......127
Hacen Dib and Mohammed Mesk
On Laguerre Polynomials of two Variables.....................................135
Samira Ibenmouloud and Mohamed Sbai
Poisson Transform on H3.......................................................................139
Hassan Sami
Transfert Formula in the Real Hyperbolic Space Bn...................145
Abderrahman Essadiq
q-Analogue of Watanabe Unitary Transform
Associated to the (/-Continuous Gegenbauer Polynomials.........155
Meryem El Beggar
Realization of a Holomorphic Discrete Series
of the Lie Group 517(1,2) as Star-Representation . 163
JOHN’S EQUATION AND THE PLANE-TO-LINE TRANSFORM
ON R3
FULTON B. GONZALEZ
1. Introduction
In this note we provide a range characterization result for the plane-to-
line transform on R3, in terms of the ultrahyperbolic equations of Fritz John
[2]. This equation was used to characterize the range of the X-ray transform
on rapidly decreasing functions on R3, and has the feature of being invariant
under the group of orientation-preserving rigid motions of R3. While this
range result is likely not new, the proof below is fairly elementary and uses
only basic facts from vector calculus, Riemannian geometry, and a classical
integral formula involving Legendre polynomials.
2. The Plane-to-Line Transform
Let M be a manifold acted on by a Lie group G. We denote the Lie
group action by (g,p) »-*» g · p. Then G also acts on the functions on M by
g · f(x) = f(g~* l - x). By differentiating this action, we obtain an action,
via vector fields, of the Lie algebra g of G on the smooth functions of M:
Z · f(x) = d/dt(f (exp(-tZ) · *))Ι*=ο, for all Z G g, / G By
composing the above vector fields, we obtain a representation of the universal
enveloping algebra u(g) on Each element of u(g) then acts as a
differential operator on M (possibly the zero operator). If U G u(g), g G G,
and / G C°°(M), then g-(U · /) = (Ad(g)U) · (g · /). Thus if G is connected
and U belongs to the center 3(g) of u(g), then U acts as a G-invariant
differential operator on M.
Let G(l,3) and G(2,3) denote, respectively, the manifolds of unoriented
lines and planes in R3. These are vector bundles over the compact Grass-
mannian manifolds Gi,s and £2,3 of lines and planes, respectively, through
the origin in R3. (Both spaces equal projective 2-space RP2.) The projec
tion of <3(1,3) onto Gi,3 maps each line l to the parallel line σ through the
origin; the fiber over each σ G G\$ can then be identified with the plane
σ1. Similarly, the projection of G(2,3) onto G2,3 maps each plane ξ onto
the parallel plane r through the origin; the fiber over r is identified with the
line rL.
Let E(3) denote the group of rigid motions of R3. E(3) equals the semidi-
rect product 0(3) xi R3; it acts transitively on both G(l,3) and G(2,3). If
σο denotes the x-axis and to the xy-plane, then the isotropy subgroups in
Date: March 30, 1999.
1991 Mathematics Subject Classification. Primary: 44A12; Secondary: 43A85.
Key words and phrases. Radon transform, Grassmannian, John’s equation.
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