[DJVU] [1984] Group Representations and Special Functions (Mathe

DOWNLOAD MORE EBOOKS, MAGAZINES, & PDFS: http://buyabook.ws/download-ebooks-magazines-pdfs-more/ BOOK INFO: Group Representations and Special Functions: Examples and Problems prepared by Aleksander Strasburger (Mathematics and its Applications) By A. Wawrzynczyk Publisher: Springer Number Of Pages: 708 Publication Date: 1984-03-31 ISBN-10 / ASIN: 9027712697 ISBN-13 / EAN: 9789027712691 Table of Contents Editor's Preface Table of Contents Preface Part I 1. Groups and Homogeneous Spaces 1.1. Groups Problems 1.2. Differentiable manifolds Problems 1.3. Lie groups and Lie algebras Problems 1.4. Transformation groups. Invariant tensor fields 1.5. Additional structures on manifolds Problems 1.6. The Hurwitz measure Problems 1.7. Quasi-invariant measures Problems 1.8. Elements of the classification of the Lie groups and algebras Problems 2. Representations of Locally Compact Groups 2.1. Definition of a representation. Examples 2.2. Basic constructions of representations. Induced representations 2.3. Further constructions of representations 2.4. Intertwining 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 3.5. The circle group and the Fourier series 3.6. Fourier analysis on a commutative locally compact group Problems 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 for matrix elements 4.6. Characters of finite-dimensional representations 4.7. Harmonic analysis on compact groups and on their homogeneous spaces Problems 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 Problems Part II 6. The Euler Γ- and B-functions 6.1. Definition of the Γ-function 6.2. The Fourier transformation and the Mellin transformation 6.3. The reflexion formula for the Γ-function 6.4. The Riemann ζ-function Problems 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 Problems 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 Plmn 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 Problems 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. The mean value theorem for a spherical function Problems 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 Uσ and the function Plmn 10.4. Differentials of the representations Uσ recurrence relations. Irreducibility 10.5. Harmonic analysis on the disc SU(1,1)/K Problems 11. 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 K 11.4. Decomposition into plane waves on K 11.5. Differential properties of spherical functions 11.6. The Gelfand-Graev transformation 11.7. Irreducibility problems of the representation Ul Problems 12. The Laguerre Polynomials 12.1. The group, the representations, 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 Problems 13. 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 Problems Part III Introduction 14. Affine Transformations 14.1. Associated vector bundles 14.2. Operations on differential forms 14.3. Affine connections 14.4. Parallel translation. Geodesics. The exponential mapping 14.5. Covariant differentiation 14.6. Affine mappings 14.7. The Riemannian connexion. Sectional curvature Problems 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 symmetric space 15.4. Geometry of symmetric spaces 15.5. Riemannian symmetric spaces. Riemann pairs 15.6. A symmetric pair is a Gelfand pair Problems 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 Problems 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 Subject Index

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