Group representations and special functions

Antoni Wawrzyńczyk ; examples and problems prepared by Aleksander Strasburger


  • I.- 1. Groups and Homogeneous Spaces.- 1.1. Groups.- 1.2. Differentiate Manifolds.- 1.3. Lie Groups and Lie Algebras.- 1.4. Transformation Groups. Invariant Tensor Fields.- 1.5. Additional Structures on Manifolds.- 1.6. The Hurwitz Measure.- 1.7. Quasi-Invariant Measures.- 1.8. Elements of the Classification of Lie Groups and Algebras.- 2. Representations of Locally Compact Groups.- 2.1. Definition of a Representation. Examples.- 2.2. Basic Constructions. Induced Representations.- 2.3. Further Constructions of Representations.- 2.4. Intertwinning Operators. Unitary Equivalence of Representations.- 2.5. Positive Definite Measures and Cyclic Representations.- 2.6. Matrix Elements of Representations.- 2.7. Group Algebra Representations and Group Representations.- 2.8. The Universal Enveloping Algebra of a Lie Group Algebra. The Differential of a Representation.- 3. Decomposition Theory of Unitary Representations.- 3.1. Irreducible Representations. Schur's Lemma.- 3.2. Classical Fourier Transformation.- 3.3. The Fourier Transforms of Functions in D (Rn).- 3.4. Analysis on the Multiplicative Group R+. The Mellin Transformation.- 3.1. The Circle Group and the Fourier Series.- 3.2. Fourier Analysis on a Commutative Locally Compact Group.- 4. Representations of Compact Groups.- 4.1. Operators of the Hilbert-Schmidt Type.- 4.2. The Tensor Product of Hilbert Spaces.- 4.3. The Frobenius Theorem.- 4.4. The Peter-Weyl Theory.- 4.5. The Orthogonality Relations of Matrix Elements.- 4.6. Characters of Finite-Dimensional Representations.- 4.7. Harmonic Analysis on Compact Groups and on Their Homogeneous Spaces.- 5. Theory of Spherical Functions.- 5.1. The Spherical Integral Equation.- 5.2. Spherical Functions and Spherical Representations.- 5.3. Existence of Spherical Functions. Gelfand Pairs.- 5.4. Differentiability of Spherical Functions on Lie Groups.- II.- 6. The Euler ?- and B-Functions.- 6.1. Definition of the ?-Function.- 6.2. The Fourier Transformation and the Mellin Transformation.- 6.3. The Reflection Formula for the ?-Function.- 6.4. The Riemann ?-Function.- 7. Bessel Functions.- 7.1. The Group of Rigid Motions of R2.- 7.2. Spherical Representations of the Group M(2).- 7.3. Properties of the Bessel Functions.- 7.4. Harmonic Analysis on the Symmetric Space of the Motion Group M(2). The Fourier-Bessel Transformation.- 8. Theory of Jacobi and Legendre Polynomials.- 8.1. Representations of the Group SL(2, C) on a Space of Polynomials.- 8.2. Properties of the Representations Tl and Their Consequences.- 8.3. Integral Equations for the Functions Pjkl.- 8.4. The Differential of the Representation Tl. Recurrence and Differential Equations for the Functions Pmnl.- 8.5. Characters of Irreducible Representations and New Integral Formulas for Legendre Functions.- 8.6. Harmonic Analysis on the Group SU(2) and the Sphere S2.- 8.7. Decomposition of the Tensor Product of Representations Tl. The Clebsch-Gordan Coefficients.- 9. Gegenbauer Polynomials.- 9.1. Information about the Group SO(n) and the Homogeneous Space Sn-1.- 9.2. Spherical Representations of the Group SO(n).- 9.3. Gegenbauer's Equation and Basic Recurrences.- 9.4. Integral Formulas for the Gegenbauer Polynomials.- 9.5. A Mean Value Theorem for a Spherical Function.- 10. Jacobi and Legendre Functions.- 10.1. Structure of the Group SL(2, R) and Its Homogeneous Spaces.- 10.2. Induced Representations of the Group SL(2,R).- 10.3. Properties of the Representation U? and the Function Bmnl.- 10.4. Differentials of the Representations U?. Recurrence Relations. Irreducibility.- 10.5. Harmonic Analysis on the Disc SU(1, 1)/K.- Chapter11. Harmonic Analysis on the Lobatschevsky space.- 11.1. The Group SL(2, C). Induced Spherical Representations.- 11.2. On the Structure of the Lobatschevsky Space.- 11.3. The Spherical Fourier Transformation on ?.- 11.4. Decomposition into Plane Waves on ?.- 11.5. Differential Properties of Spherical Functions.- 11.6. The Gelfand-Graev Transformation.- 11.7. Irreducibility Problems of the Representations Ul.- 12. The Laguerre Polynomials.- 12.1. The Group, the Representation, Matrix Elements.- 12.2. Basic Properties of the Laguerre Polynomials.- 12.3. Differential Properties of the Laguerre Polynomials.- 12.4. One-Dimensional Harmonic Oscillator and the Hermite Polynomials.- 12.5. Connection between the Laguerre Polynomials and the Jacobi Functions.- 12.6. Orthogonality Relations for the Laguerre Polynomials.- Chapter13. The Hypergeometric Equation.- 13.1. The Second Order Homogeneous Linear Differential Equation on C.- 13.2. Solutions of the Hypergeometric Equation in the Form of Euler Integrals.- 13.3. The Hypergeometric Function for Some Special Values of the Parameters.- 13.4. The Confluent Hypergeometric Equation and the Confluent Hypergeometric Function.- III.- 14. Affine Transformations.- 14.1. Associated Vector Bundles.- 14.2. Operations on Differential Forms.- 14.3. Affine Connections.- 14.4. Parallel Translation. Geodesies. The Exponential Mapping.- 14.5. Covariant Differentiation.- 14.6. Affine Mappings.- 14.7. The Riemannian Connection. Sectional Curvature.- 15. Symmetric Spaces.- 15.1. Definitions and Examples.- 15.2. Affine Connection on a Symmetric Space.- 15.3. Structure of the Group of Displacements of a Symetric Space.- 15.4. Geometry of Symmetric Spaces.- 15.5. Riemannian Symmetric Spaces. Riemann Pairs.- 15.6. A Symmetric Pair is a Gelfand Pair.- 16. General Harmonic Analysis on a Symmetric Space.- 17. Semisimple Algebras. Semisimple Groups. Symmetric Spaces of the Non-Compact Type.- 17.1. Compact Lie Algebras.- 17.2. Structure of Semisimple Algebras.- 17.3. Iwasawa Decomposition of an Algebra and of a Group.- 17.4. The Weyl Group.- 17.5. Boundary of a Symmetric Space of the Non-Compact Type.- 17.6. Planes and Horocycles in a Symmetric Space.- 18. Harmonic Analysis on Symmetric Spaces of the Non-Compact Type.- 18.1. Plane Waves and Spherical Functions.- 18.2. The Fourier Transformation on a Symmetric Space.- 18.3. Properties of Spherical Functions.- 18.4. Asymptotic Behaviour of a Spherical Function. The Harish-Chandra c(*)-Function.- 18.5. Properties of the Harish-Chandra c(*)-Function.- 18.6. The Plancherel Formula for the Fourier transformation on a Symmetric Space.- 18.7. The Radon Transformation.- 18.8. The Paley-Wiener Theorem.- Table of Formulas.- References.- List of Symbols.- Author Index.

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書名 Group representations and special functions
著作者等 Strasburger, Aleksander
Wawrzyńczyk, Antoni
Ziemian B.
Wawrzynczyk A.
書名別名 Współczena teoria funkcji specjalnych
シリーズ名 Mathematics and its applications
出版元 D.Reidel
Distributed for the U.S.A. and Canada, Kluwer Boston
PWN-Polish Scientific
刊行年月 c1984
ページ数 xvi, 688 p.
大きさ 23 cm
ISBN 9027712697
NCID BA03468250
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言語 英語
原文言語 ポーランド語
出版国 オランダ