Groups & Geometric Analysis
Radon Transforms, Invariant Differential Operators and Spherical Functions: Volume 1
Published on 30. June 1984
654 pages
978-0-08-087432-6 (ISBN)
Description
Groups & Geometric Analysis
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Sigurdur Helgason
Group and Geometric Analysis
Integral Geometry, Invariant Differential Operators and Spherical Functions
Book
09/1997
Academic Press
€55.71
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Content
- Front Cover
- Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
- Copyright Page
- Contents
- Preface
- Suggestions to the Reader
- Tentative Contents of the Sequel
- Introduction: Geometric Fourier Analysis on Spaces of Constant Curvature
- 1. Harmonic Analysis on Homogeneous Spaces
- 2. The Euclidean Plane R2
- 3. The Sphere S2
- 4. The Hyperbolic Plane H2
- Chapter I. Integral Geometry and Radon Transforms
- 1. Integration on Manifolds
- 2. The Radon Transform on Rn
- 3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
- 4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform
- 5. Integral Formulas
- 6. Orbital Integrals
- Exercises and Further Results
- Notes
- Chapter II. Invariant Differential Operators
- 1. Differentiable Functions on Rn
- 2. Differential Operators on Manifolds
- 3. Geometric Operations on Differential Operators
- 4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces
- 5. Invariant Differential Operators on Symmetric Spaces
- Exercises and Further Results
- Notes
- Chapter III. Invariants and Harmonic Polynomials
- 1. Decomposition of the Symmetric Algebra. Harmonic Polynomials
- 2. Decomposition of the Exterior Algebra. Primitive Forms
- 3. Invariants for the Weyl Group
- 4. The Orbit Structure of p
- 5. Harmonic Polynomials on p
- Exercises and Further Results
- Notes
- Chapter IV. Spherical Functions and Spherical Transforms
- 1. Representations
- 2. Spherical Functions: Preliminaries
- 3. Elementary Properties of Spherical Functions
- 4. Integral Formulas for Spherical Functions. Connections with Representations
- 5. Harish-Chandra's Spherical Function Expansion
- 6. The c-Function
- 7. The Paley-Wiener Theorem and the Inversion Formula for the Spherical Transform
- 8. The Bounded Spherical Functions
- 9. The Spherical Transform on p, the Euclidean Type
- 10. Convexity Theorems
- Exercises and Further Results
- Notes
- Chapter V. Analysis on Compact Symmetric Spaces
- 1. Representations of Compact Lie Groups
- 2. Fourier Expansions on Compact Groups
- 3. Fourier Decomposition of a Representation
- 4. The Case of a Compact Symmetric Space
- Exercises and Further Results
- Notes
- Solutions to Exercises
- Appendix
- 1. The Finite-Dimensional Representations of sl(2, C)
- 2. Representations and Reductive Lie Algebras
- 3. Some Algebraic Tools
- Bibliography
- Symbols Frequently Used
- Index
