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" Groups and geometric analysis : "
Sigurdur Helgason.
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BL
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| Record Number
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637141
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dltt
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| Main Entry
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Helgason, Sigurdur,1927-
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| Title & Author
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Groups and geometric analysis : : integral geometry, invariant differential operators, and spherical functions /\ Sigurdur Helgason.
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| Publication Statement
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Providence, R.I. :: American Mathematical Society,, 2000.
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| Series Statement
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Mathematical surveys and monographs ;; v. 83
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| Page. NO
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xxii, 667 p. :: ill. ;; 26 cm.
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| ISBN
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0821826735 (alk. paper)
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: 9780821826737 (alk. paper)
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| Notes
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Originally published: Orlando : Academic Press, c1984.
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| Bibliographies/Indexes
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Includes bibliographical references (p. 619-653) and index.
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| Contents
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Introduction: Geometric Fourier Analysis on Spaces of Constant Curvature -- 1. Harmonic Analysis on Homogeneous Spaces 1 -- 2. Notation and Preliminaries 2 -- 2. The Euclidean Plane R[superscript 2] 4 -- 1. Eigenfunctions and Eigenspace Representations 4 -- 2. A Theorem of Paley -- Wiener Type 15 -- 3. The Sphere S[superscript 2] 16 -- 1. Spherical Harmonics 16 -- 2. Proof of Theorem 2.10 23 -- 4. The Hyperbolic Plane H[superscript 2] 29 -- 1. Non-Euclidean Fourier Analysis. Problems and Results 29 -- 2. The Spherical Functions and Spherical Transforms 38 -- 3. The Non-Euclidean Fourier Transform. Proof of the Main Result 44 -- 4. Eigenfunctions and Eigenspace Representations. Proofs of Theorems 4.3 and 4.4 58 -- 5. Limit Theorems 69 -- Chapter 1 Integral Geometry and Radon Transforms -- 1. Integration on Manifolds 81 -- 1. Integration of Forms. Riemannian Measure 81 -- 2. Invariant Measures on Coset Spaces 85 -- 3. Haar Measure in Canonical Coordinates 96 -- 2. The Radon Transform on R[superscript n] 96 -- 2. The Radon Transform of the Spaces D(R[superscript n]) and S(R[superscript n]). The Support Theorem 97 -- 3. The Inversion Formulas 110 -- 4. The Plancherel Formula 115 -- 5. The Radon Transform of Distributions 117 -- 6. Integration over d-Planes. X-Ray Transforms 122 -- 7. Applications 126 -- A. Partial Differential Equations 126 -- B. Radiography 130 -- 8. Appendix. Distributions and Riesz Potentials 131 -- 3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals 139 -- 1. A Duality for Homogeneous Spaces 139 -- 2. The Radon Transform for the Double Fibration 143 -- 3. Orbital Integrals 149 -- 4. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform 150 -- 1. Spaces of Constant Curvature 151 -- A. The Hyperbolic Space 152 -- B. The Spheres and the Elliptic Spaces 161 -- 2. Compact Two-Point Homogeneous Spaces 164 -- 3. Noncompact Two-Point Homogeneous Spaces 177 -- 4. The X-Ray Transform on a Symmetric Space 178 -- 5. Integral Formulas 180 -- 1. Integral Formulas Related to the Iwasawa Decomposition 181 -- 2. Integral Formulas for the Cartan Decomposition 186 -- A. The Noncompact Case 186 -- B. The Compact Case 187 -- C. The Lie Algebra Case 195 -- 3. Integral Formulas for the Bruhat Decomposition 196 -- 6. Orbital Integrals 199 -- 1. Pseudo-Riemannian Manifolds of Constant Curvature 199 -- 2. Orbital Integrals for the Lorentzian Case 203 -- 3. Generalized Riesz Potentials 211 -- 4. Determination of a Function from Its Integrals over Lorentzian Spheres 214 -- 5. Orbital Integrals on SL(2,R) 218 -- Chapter II Invariant Differential Operators -- 1. Differentiable Functions on R[superscript n] 233 -- 2. Differential Operators on Manifolds 239 -- 1. Definition. The Spaces D(M) and E(M) 239 -- 2. Topology of the Spaces D(M) and E(M). Distributions 239 -- 3. Effect of Mappings. The Adjoint 241 -- 4. The Laplace-Beltrami Operator 242 -- 3. Geometric Operations on Differential Operators 251 -- 1. Projections of Differential Operators 251 -- 2. Transversal Parts and Separation of Variables for Differential Operators 253 -- 3. Radial Parts of a Differential Operator. General Theory 259 -- 4. Examples of Radial Parts 265 -- 4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces 274 -- 2. The Algebra D(G/H) 280 -- 3. The Case of a Two-Point Homogeneous Space. The Generalized Darboux Equation 287 -- 5. Invariant Differential Operators on Symmetric Spaces 289 -- 1. The Action on Distributions and Commutativity 289 -- 2. The Connection with Weyl Group Invariants 295 -- 3. The Polar Coordinate Form of the Laplacian 309 -- 4. The Laplace-Beltrami Operator for a Symmetric Space of Rank One 312 -- 5. The Poisson Equation Generalized 315 -- 6. Asgeirsson's Mean-Value Theorem Generalized 318 -- 7. Restriction of the Central Operators in D(G) 323 -- 8. Invariant Differential Operators for Complex Semisimple Lie Algebras 326 -- 9. Invariant Differential Operators for X = G/K, G Complex 329 -- Chapter III Invariants and Harmonic Polynomials -- 1. Decomposition of the Symmetric Algebra. Harmonic Polynomials 345 -- 2. Decomposition of the Exterior Algebra. Primitive Forms 354 -- 3. Invariants for the Weyl Group 356 -- 1. Symmetric Invariants 356 -- 2. Harmonic Polynomials 360 -- 3. The Exterior Invariants 363 -- 4. Eigenfunctions of Weyl Group Invariant Operators 364 -- 5. Restriction Properties 366 -- 4. The Orbit Structure of p 368 -- 2. Nilpotent Elements 370 -- 3. Regular Elements 373 -- 4. Semisimple Elements 378 -- 5. Algebro-Geometric Results on the Orbits 380 -- 5. Harmonic Polynomials on p 380 -- Chapter IV Spherical Functions and Spherical Transforms -- 1. Representations 385
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| Subject
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Lie groups.
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Geometry, Differential.
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| LC Classification
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QA387.H45 2000
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