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Tables of Contents for The Radon Transform
Chapter/Section Title
Page #
Page Count
Preface to the Second Edition
ix
2
Preface to the First Edition
xi
CHAPTER I The Radon Transform on R^n
1
52
1. Introduction
1
1
2. The Radon Transform of the Spaces D(R^n) and S(R^n). The Support Theorem
2
13
3. The Inversion Formula
15
5
4. The Plancherel Formula
20
2
5. Radon Transform of Distributions
22
6
6. Integration over d-Planes. X-ray Transforms. The Range of the d-Plane Transform
28
13
7. Applications
41
10
a) Partial Differential Equations
41
5
b) X-ray Reconstruction
46
5
Bibliographical Notes
51
2
CHAPTER II A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
53
30
1. Homogeneous Spaces in Duality
53
4
2. The Radon Transform for the Double Fibration
57
5
3. Orbital Integrals
62
1
4. Examples of Radon Transforms for Homogeneous Spaces in Duality
63
17
A. The Funk Transform
63
3
B. The X-ray Transform in H^2
66
1
C. The Horocycles in H^2
67
4
D. The Poisson Integral as a Radon Transform
71
2
E. The d-Plane Transform
73
1
F. Grassmann Manifolds
74
1
G. Half-lines in a Half-plane
75
4
H. Theta Series and Cusp Forms
79
1
Bibliographical Notes
80
3
CHAPTER III The Radon Transform on Two-Point Homogeneous Spaces
83
40
1. Spaces of Constant Curvature. Inversion and Support Theorems
83
28
A. The Hyperbolic Space
85
8
B. The Spheres and the Elliptic Spaces
93
15
C. The Spherical Slice Transform
108
3
2. Compact Two-Point Homogeneous Spaces. Applications
111
7
3. Noncompact Two-Point Homogeneous Spaces
118
1
4. The X-ray Transform on a Symmetric Space
119
1
5. Maximal Tori and Minimal Spheres in Compact Symmetric Spaces
120
2
Bibliographical Notes
122
1
CHAPTER IV Orbital Integrals and the Wave Operator for Isotropic Lorentz Spaces
123
24
1. Isotropic Spaces
123
5
A. The Riemannian Case
124
1
B. The General Pseudo-Riemannian Case
124
4
C. The Lorentzian Case
128
1
2. Orbital Integrals
128
9
3. Generalized Riesz Potentials
137
3
4. Determination of a Function from Its Integrals over Lorentzian Spheres
140
4
5. Orbital Integrals and Huygens' Principle
144
1
Bibliographical Notes
145
2
CHAPTER V Fourier Transforms and Distributions. A Rapid Course
147
24
1. The Topology of the Spaces D(R^n), SIGMA(R^n) and S(R^n)
147
2
2. Distributions
149
1
3. The Fourier Transform
150
6
4. Differential Operators with Constant Coefficients
156
4
5. Riesz Potentials
160
8
Bibliographical Notes
168
3
Bibliography
171
14
Notational Conventions
185
2
Subject Index
187