Group Representations and Special Functions: Examples and Problems Prepared by Aleksander Strasburger

ISBN-13: 9789027722942 / Angielski / Miękka / 1986 / 688 str.

A. Wawrzynczyk

Opis

Growing specialization and diversification have brought a hor'st of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are sudden- ly seen to be related. Further, the kind and level of sophistication of mathematics applied invarious sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This programme, Mathematics and Its Applications, is devoted to such (new) interrelations as exempli gratia: - a central concept which plays an important role in several different mathematical andjor scientific specialized areas; - new applications of the results and ideas from one area of scien- tific endeavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.

Recenzje

`...the author's presentation is attractive and lucid, quite suitable for a graduate level course on spherical functions with applications to special function.'
Bulletin of the American Mathematical Society, July 1985

Spis treści

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.