Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations / by Audrey Terras.

Format:

Book

Author/Creator:

Terras, Audrey., Author.

Language:

English

Subjects (All):

Harmonic analysis.

Number theory.

Geometry.

Discrete mathematics.

Mathematics.

Statistics.

Abstract Harmonic Analysis.

Number Theory.

Discrete Mathematics.

Applications of Mathematics.

Statistical Theory and Methods.

Local Subjects:

Abstract Harmonic Analysis.

Number Theory.

Geometry.

Discrete Mathematics.

Applications of Mathematics.

Statistical Theory and Methods.

Physical Description:

1 online resource (XV, 487 p. 41 illus., 21 illus. in color.)

Edition:

2nd ed. 2016.

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 2016.

Language Note:

English

Summary:

This text explores the geometry and analysis of higher rank analogues of the symmetric spaces introduced in volume one. To illuminate both the parallels and differences of the higher rank theory, the space of positive matrices is treated in a manner mirroring that of the upper-half space in volume one. This concrete example furnishes motivation for the general theory of noncompact symmetric spaces, which is outlined in the final chapter. The book emphasizes motivation and comprehensibility, concrete examples and explicit computations (by pen and paper, and by computer), history, and, above all, applications in mathematics, statistics, physics, and engineering. The second edition includes new sections on Donald St. P. Richards’s central limit theorem for O(n)-invariant random variables on the symmetric space of GL(n, R), on random matrix theory, and on advances in the theory of automorphic forms on arithmetic groups.

Contents:

Intro

Preface to the First Edition

Preface to the Second Edition

Contents

List of Figures

1 The Space Pn of Positive nn Matrices

1.1 Geometry and Analysis on Pn

1.1.1 Introduction

1.1.2 Elementary Results

1.1.3 Geodesics and Arc Length

1.1.4 Measure and Integration on Pn

1.1.5 Differential Operators on Pn

1.1.6 A List of the Main Formulas Derived in Section 1.1

1.1.7 An Application to Multivariate Statistics

1.2 Special Functions on Pn

1.2.1 Power and Gamma Functions

1.2.2 K-Bessel Functions

1.2.3 Spherical Functions

1.2.4 The Wishart Distribution

1.2.5 Richards' Extension of the Asymptotics of Spherical Functions for P3 to Pn for General n

1.3 Harmonic Analysis on Pn in Polar Coordinates

1.3.1 Properties of the Helgason-Fourier Transform on Pn

1.3.2 Beginning of the Discussion of Part (1) of Theorem 1.3.1-Steps 1 and 2

1.3.3 End of the Discussion of Part (1) of Theorem 1.3.1-Steps 3 and 4

1.3.4 Applications-Richards' Central Limit Theorem for K-Invariant Functions on Pn

1.3.5 Quantum Chaos and Random Matrix Theory

1.3.6 Other Directions in the Labyrinth

1.4 Fundamental Domains for Pn/GL(n,Z)

1.4.1 Introduction

1.4.2 Minkowski's Fundamental Domain

1.4.3 Grenier's Fundamental Domain

Grenier's Reduction Algorithm

1.4.4 Integration over Fundamental Domains

1.5 Maass Forms for GL(n,Z) and Harmonic Analysis on Pn/GL(n,Z)

1.5.1 Analytic Continuation of Eisenstein Series by the Method of Inserting Larger Parabolic Subgroups

1.5.2 Hecke Operators and Analytic Continuation of L-Functions Associated with Maass Forms by the Method of Theta Functions

1.5.3 Fourier Expansions of Eisenstein Series

Generalities on Fourier Expansions of Eisenstein Series

Remarks on Maass Cusp Forms.

1.5.4 Update on Maass Cusp Forms for SL(3,Z) and L-Functions Plus Truncating Eisenstein Series

Maass Cusp Forms for SL(3,Z) and L-Functions

Langlands' Inner Product Formulas for Truncated Eisenstein Series

1.5.5 Remarks on Harmonic Analysis on the Fundamental Domain

1.5.6 Finite and Other Analogues

2 The General Noncompact Symmetric Space

2.1 Geometry and Analysis on G/K

2.1.1 Symmetric Spaces, Lie Groups, and Lie Algebras

2.1.2 Examples of Symmetric Spaces

Plan for Construction of Noncompact Symmetric Spaces of Type III

Type a Examples

Type c Examples

2.1.3 Cartan, Iwasawa, and Polar Decompositions, Roots

Three Examples of Iwasawa Decompositions of Real Semisimple Lie Algebras

Examples of the Polar Decomposition

2.1.4 Geodesics and the Weyl Group

2.1.5 Integral Formulas

Examples

Invariant Volume Elements on the Symmetric Spaces of GL(n,R) and Sp(n,R)

2.1.6 Invariant Differential Operators

2.1.7 Special Functions and Harmonic Analysis on Symmetric Spaces

2.1.8 An Example of a Symmetric Space of Type IV: The Quaternionic Upper Half 3-Space

2.2 Geometry and Analysis on "026E30F G/K

2.2.1 Fundamental Domains

2.2.2 Automorphic Forms

Questions Arising from Proposition 2.2.3

2.2.3 Trace Formulas

Trace Formula for Discrete Acting on the Quaternionic Upper Half Plane

Trace Formula for Discrete Acting on Hm

Trace Formula for Acting on the Siegel Upper Half Space

References

Index.

Notes:

Bibliographic Level Mode of Issuance: Monograph

ISBN:

1-4939-3408-2

OCLC:

948661162