Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions / Sigurdur Helgason.

Format:

Book

Author/Creator:

Helgason, Sigurdur, 1927- author.

Series:

Mathematical surveys and monographs ; no. 83.

Mathematical surveys and monographs ; volume 83

Language:

English

Subjects (All):

Lie groups.

Geometry, Differential.

Physical Description:

1 online resource (693 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2000.

Language Note:

English

Summary:

Group-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group. The introductory chapter is a self-contained account of the analysis on surfaces of constant curvature. Later chapters cover general cases of the Radon transform, spherical functions, invariant operators, compact symmetric spaces and other topics. This book, together with its companion volume, Geometric Analysis on Symmetric Spaces (AMS Mathematical Surveys and Monographs series, vol. 39, 1994), has become the standard text for this approach to geometric analysis.Sigurdur Helgason was awarded the Steele Prize for outstanding mathematical exposition for Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces.

Contents:

""CONTENTS""; ""PREFACE""; ""PREFACE TO THE 2000 PRINTING""; ""SUGGESTIONS TO THE READER""; ""A SEQUEL TO THE PRESENT VOLUME""; ""INTRODUCTION: GEOMETRIC FOURIER ANALYSIS ON SPACES OF CONSTANT CURVATURE""; ""1. Harmonic Analysis on Homogeneous Spaces""; ""1. General Problems""; ""2. Notation and Preliminaries""; ""2. The Euclidean Plane R[sup(2)]""; ""1. Eigenfunctions and Eigenspace Representations""; ""2. A Theorem of Paley-Wiener Type""; ""3. The Sphere S[sup(2)]""; ""1. Spherical Harmonics""; ""2. Proof of Theorem 2.10""; ""4. The Hyperbolic Plane H[sup(2)]""

""1. Non-Euclidean Fourier Analysis. Problems and Results""""2. The Spherical Functions and Spherical Transforms""; ""3. The Non-Euclidean Fourier Transform. Proof of the Main Result""; ""4. Eigenfunctions and Eigenspace Representations. Proofs of Theorems 4.3 and 4.4""; ""5. Limit Theorems""; ""Exercises and Further Results""; ""Notes""; ""CHAPTER I: INTEGRAL GEOMETRY AND RADON TRANSFORMS""; ""1. Integration on Manifolds""; ""1. Integration of Forms. Riemannian Measure""; ""2. Invariant Measures on Coset Spaces""; ""3. Haar Measure in Canonical Coordinates""

""2. The Radon Transform on R[sup(n)]""""1. Introduction""; ""2. The Radon Transform of the Spaces D(R[sup(n)]) and p(R[sup(n)]). The Support Theorem""; ""3. The Inversion Formulas""; ""4. The Plancherel Formula""; ""5. The Radon Transform of Distributions""; ""6. Integration over d-Planes. X-Ray Transforms""; ""7. Applications""; ""A. Partial Differential Equations""; ""B. Radiography""; ""8. Appendix. Distributions and Riesz Potentials""; ""3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals""; ""1. A Duality for Homogeneous Spaces""

""2. The Radon Transform for the Double Fibration""""3. Orbital Integrals""; ""4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform""; ""1. Spaces of Constant Curvature""; ""A. The Hyperbolic Space""; ""B. The Spheres and the Elliptic Spaces""; ""2. Compact Two-Point Homogeneous Spaces""; ""3. Noncompact Two-Point Homogeneous Spaces""; ""4. The X-Ray Transform on a Symmetric Space""; ""5. Integral Formulas""; ""1. Integral Formulas Related to the Iwasawa Decomposition""; ""2. Integral Formulas for the Cartan Decomposition""; ""A. The Noncompact Case""

""B. The Compact Case""""C. The Lie Algebra Case""; ""3. Integral Formulas for the Bruhat Decomposition""; ""6. Orbital Integrals""; ""1. Pseudo-Riemannian Manifolds of Constant Curvature""; ""2. Orbital Integrals for the Lorentzian Case""; ""3. Generalized Riesz Potentials""; ""4. Determination of a Function from Its Integrals over Lorentzian Spheres""; ""5. Orbital Integrals on SL(2,R)""; ""Exercises and Further Results""; ""Notes""; ""CHAPTER II: INVARIANT DIFFERENTIAL OPERATORS""; ""1. Differentiable Functions on R[sup(n)]""; ""2. Differential Operators on Manifolds""

""1. Definition. The Spaces D(M) and E(M)""

Notes:

Originally published: Orlando : Academic Press, c1984.

Includes bibliographical references (pages 619-653) and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1310-8

OCLC:

922980122