**Contemporary Mathematics**

Volume: 598;
2013;
280 pp;
Softcover

MSC: Primary 22; 43; 44; 45; 92;
Secondary 32; 35; 65

Print ISBN: 978-0-8218-8738-7

Product Code: CONM/598

List Price: $100.00

Individual Member Price: $80.00

**Electronic ISBN: 978-1-4704-1026-1
Product Code: CONM/598.E**

List Price: $100.00

Individual Member Price: $80.00

# Geometric Analysis and Integral Geometry

Share this page *Edited by *
*Eric Todd Quinto; Fulton Gonzalez; Jens Gerlach Christensen*

This volume contains the proceedings of the AMS Special Session on
Radon Transforms and Geometric Analysis, in honor of Sigurdur
Helgason's 85th Birthday, held from January 4–7, 2012, in Boston, MA,
and the Tufts University Workshop on Geometric Analysis on Euclidean
and Homogeneous Spaces, held from January 8–9, 2012, in Medford, MA.

This volume provides an historical overview of several decades in
integral geometry and geometric analysis as well as recent advances in
these fields and closely related areas. It contains several articles
focusing on the mathematical work of Sigurdur Helgason, including an
overview of his research by Gestur Ólafsson and Robert Stanton.
The first article in the volume contains Helgason's own reminiscences
about the development of the group-theoretical aspects of the Radon
transform and its relation to geometric analysis. Other contributions
cover Radon transforms, harmonic analysis, Penrose transforms,
representation theory, wavelets, partial differential operators on
groups, and inverse problems in tomography and cloaking that are
related to integral geometry.

Many articles contain both an overview of their respective fields as
well as new research results. The volume will therefore appeal to
experienced researchers as well as a younger generation of
mathematicians. With a good blend of pure and applied topics the
volume will be a valuable source for interdisciplinary research.

#### Table of Contents

# Table of Contents

## Geometric Analysis and Integral Geometry

- Preface ix10 free
- List of Presenters xi12 free
- Historical Articles 116 free
- Some personal remarks on the Radon transform 318
- On the Life and Work of S. Helgason 2136
- Research and Expository Articles 4358
- Microlocal analysis of an ultrasound transform with circular source and receiver trajectories 4560
- Cuspidal discrete series for projective hyperbolic spaces 5974
- 1. Introduction 5974
- 2. Model and structure 6075
- 3. The discrete series 6378
- 4. Schwartz functions 6479
- 5. A Radon transform and an Abel transform 6479
- 6. The main result 6681
- 7. Proof of the main theorem for 𝑝≥𝑞 6681
- 8. Proof of (i) - (v) of the main theorem for 𝑞>𝑝 6782
- 9. Reduction to the real case (𝑑=1) 6984
- 10. Proof of (vi) of the main theorem - A Closer study of 𝑅𝑓 near +∞ 7085
- 11. Final Remarks - the remaining cases 7388
- References 7489

- The Radon transform on 𝑆𝑂(3): motivations, generalizations, discretization 7792
- 1. Introduction 7792
- 2. Texture goniometry 7893
- 3. Fourier Analysis on compact groups 8095
- 4. Problem 5: Radon transform on compact groups 8196
- 5. Problem 2: Radon transform on 𝑆𝑂(3) 8499
- 6. Radon transforms on the group 𝑆𝑂(3) and the sphere 𝑆³ 86101
- 7. Problem 1: Inversion of crystallographic 𝑋-ray transform 90105
- 8. Problem 4: Exact reconstruction of a bandlimited function 𝑓 on 𝑆𝑂(3) from a finite number of samples of ℛ𝒻 91106
- 9. Appendix 93108
- Acknowledgment 94109
- References 94109

- Atomic decompositions of Besov spaces related to symmetric cones 97112
- A double fibration transform for complex projective space 111126
- Magnetic Schrödinger equation on compact symmetric spaces and the geodesic Radon transform of one forms 129144
- 1. Introduction 129144
- 2. Magnetic Schödinger equation on compact symmetric spaces 130145
- 3. Geodesic Radon transform of 1-forms 130145
- 4. Magnetic Schrödinger equation on real projective spaces 132147
- 5. Magnetic Schrödinger equation on spheres 134149
- 6. Some remarks 136151
- Acknowledgments 137152
- References 137152

- 𝐹-method for constructing equivariant differential operators 139154
- Schiffer’s conjecture, interior transmission eigenvalues and invisibility cloaking: Singular problem vs. nonsingular problem 147162
- Approximate Reconstruction from Circular and Spherical Mean Radon Transform Data 155170
- Analytic and group-theoretic aspects of the Cosine transform 167182
- Quantization of linear algebra and its application to integral geometry 189204
- 1. Introduction 189204
- 2. Conjugacy classes and scalar generalized Verma modules 190205
- 3. Eigenvalues and determinants 192207
- 4. Restriction to the diagonal part and completely integrable quantum systems 194209
- 5. Minors 195210
- 6. Elementary divisors 196211
- 7. Characteristic polynomials and minimal polynomials 197212
- 8. Integral geometry —Poisson transform and Penrose transform 199214
- 9. Integral geometry —Radon transform, hypergeometric functions 202217
- 10. Whittaker vectors 205220
- References 206221

- Mean value theorems on symmetric spaces 209224
- Semyanistyi fractional integrals and Radon transforms 221236
- Radon–Penrose transform between symmetric spaces 239254
- Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations 257272

#### Readership

Graduate students and researchers interested in integral geometry, representation theory, harmonic and geometric analysis.