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Group Representations and Special Functions: Examples and Problems prepared by Aleksander Strasburger - nieuw boek

ISBN: 9789027712691

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 relat… Meer...

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Group Representations and Special Functions - gebonden uitgave, pocketboek

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Group Representations and Special Functions - gebonden uitgave, pocketboek

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

The approach to the theory of special functions presented in this book is based on the theory of group representations. This stresses and reveals connections of special functions with harmonic analysis on homogeneous spaces.

The book consists of three parts. The first introduces basic material on the theory of Lie groups and their representations. The second (and largest part) is devoted to the detailed study of particular classes of special functions, including those of Jacobi, Bessel, Gegenbauer, and Legendre, Laguerre polynomials, hypergeometric, gamma, and beta functions, and others.

The third part explores in more detail subjects touched on in the preceding two sections. It includes a brief, but complete, discussion of the geometry of general symmetric spaces and the foundations of harmonic analysis on these spaces. This leads the reader naturally to the elements of the work of Harish-Chandra on the analysis of spherical functions. A large collection of examples with solutions, and problems for individual solution, are included.

The book also presents tables of basic formulae and diagrams showing the group theoretic schemes of approaching concrete classes of special functions.

Gedetalleerde informatie over het boek. - Group Representations and Special Functions: Examples and Problems prepared by Aleksander Strasburger


EAN (ISBN-13): 9789027712691
ISBN (ISBN-10): 9027712697
Gebonden uitgave
pocket book
Verschijningsjaar: 1984
Uitgever: Kluwer Academic Publishers
708 Bladzijden
Gewicht: 1,210 kg
Taal: eng/Englisch

Boek bevindt zich in het datenbestand sinds 2007-06-12T10:47:51+02:00 (Amsterdam)
Detailpagina laatst gewijzigd op 2021-12-06T18:54:19+01:00 (Amsterdam)
ISBN/EAN: 9027712697

ISBN - alternatieve schrijfwijzen:
90-277-1269-7, 978-90-277-1269-1


Gegevens van de uitgever

Auteur: A. Wawrzynczyk
Titel: Mathematics and its Applications; Group Representations and Special Functions - East European Series; Examples and Problems prepared by Aleksander Strasburger
Uitgeverij: Springer; Springer Netherland
688 Bladzijden
Verschijningsjaar: 1984-03-31
Dordrecht; NL
Gewicht: 2,560 kg
Taal: Engels
117,69 € (DE)
120,99 € (AT)
130,00 CHF (CH)
POD

BB; Book; Hardcover, Softcover / Mathematik/Arithmetik, Algebra; Gruppen und Gruppentheorie; Verstehen; B; Group Theory and Generalizations; Group Theory and Generalizations; Mathematics and Statistics; BC; EA

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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