Gel'Fand / Graev / Vilenkin Integral Geometry and Representation Theory

1. Auflage 2014
ISBN: 978-1-4832-6225-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 468 Seiten, Web PDF

Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one. This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which transformations are performed not on the points of a space, but on the functions defined on it. The topics discussed include Radon transform on a real affine space, integral transforms in the complex domain, and representations of the group of complex unimodular matrices in two dimensions. The properties of the Fourier transform on G, integral geometry in a space of constant curvature, harmonic analysis on spaces homogeneous with respect to the Lorentz Group, and invariance under translation and dilation are also described. This volume is suitable for mathematicians, specialists, and students learning integral geometry and representation theory.

Autoren/Hrsg.

Gel'Fand, I. M.

Graev, M. I.

Vilenkin, N. Ya.

Inhaltsverzeichnis

1;Front Cover;1
2;Integral Geometry and Representation Theory;4
3;Copyright Page;5
4;Table of Contents;10
5;Translator's Note;6
6;Foreword;8
7;CHAPTER I. RADON TRANSFORM OF TEST FUNCTIONS AND GENERALIZED FUNCTIONS
ON A REAL AFFINE SPACE;20
7.1;1. The Radon Transform on a Real Affine Space;20
7.2;2. The Radon Transform of Generalized Functions;40
7.3;3. Radon Transforms of Some Particular Generalized Functions;74
7.4;4. Summary of Radon Transform Formulas;88
8;CHAPTER II. INTEGRAL TRANSFORMS IN THE
COMPLEX DOMAIN;94
8.1;1. Line Complexes in a Space of Three Complex Dimensions
and Related Integral Transforms;96
8.2;2. Integral Geometry on a Quadratic Surface in a Space of
Four Complex Dimensions;113
8.3;3. The Radon Transform in the Complex Domain;134
9;CHAPTER III. REPRESENTATIONS OF THE GROUP OF COMPLEX UNIMODULAR MATRICES
IN TWO DIMENSIONS;152
9.1;1. The Group of Complex Unimodular Matrices
in Two Dimensions and Some of Its Realizations;153
9.2;2. Representations of the Lorentz Group
Acting on Homogeneous Functions of Two Complex Variables;158
9.3;3. Summary of Basic Results concerning Representations on Dx;167
9.4;4. Invariant Bilinear Functionals;176
9.5;5. Equivalence of Representations of G;197
9.6;6. Unitary Representations of G;208
10;CHAPTER IV. HARMONIC ANALYSIS ON THE GROUP OF COMPLEX UNIMODULAR MATRICES
IN TWO DIMENSIONS;221
10.1;1. Definition of the Fourier Transform on a Group.
Statement of the Problems and Summary of the Results;221
10.2;2, Properties of the Fourier Transform on G;235
10.3;3. Inverse Fourier Transform and Plancherel's Theorem for G;246
10.4;4. Differential Operators on G;266
10.5;5. The Paley-Wiener Theorem for the Fourier Transform on G;275
11;CHAPTER V. INTEGRAL GEOMETRY
IN A SPACE OF CONSTANT CURVATURE;292
11.1;1. Spaces of Constant
Curvature;293
11.2;2. Integral Transform Associated with Horospheres
in a Lobachevskian Space;309
11.3;3. Integral Transform Associated with Horospheres
in an Imaginary Lobachevskian Space;323
12;CHAPTER VI. HARMONIC ANALYSIS ON SPACES HOMOGENEOUS WITH RESPECT TO THE
LORENTZ GROUP;350
12.1;1. Homogeneous Spaces and the Associated Representations
of the Lorentz Group;350
12.2;2. Representations of the Lorentz Group Associated with the Complex Affine Plane and with the Cone,
and Their Irreducible Components;368
12.3;3. Decomposition of the Representation of the Lorentz Group
Associated with Lobachevskian Space;375
12.4;4. Decomposition of the Representation of the Lorentz Group
Associated with Imaginary Lobachevskian Space;383
12.5;5. Integral Geometry and Harmonic Analysis on the
Point Pairs on the Complex Projective Line;404
13;CHAPTER VII. REPRESENTATIONS OF THE GROUP OF REAL UNIMODULAR MATRICES
IN TWO DIMENSIONS;409
13.1;1. Representations of the Real Unimodular Matrices in Two Dimensions Acting on Homogeneous Functions
of Two Real Variables;409
13.2;2. Summary of the Basic Results concerning
Representations on Dx;414
13.3;3. Invariant Bilinear Functionals;419
13.4;4. Equivalence of Two Representations;432
13.5;5. Unitary Representations of G;443
14;NOTES AND REFERENCES
TO THE LITERATURE;459
15;BIBLIOGRAPHY;461
16;Index;464