Harmonic analysis on symmetric spaces-- Euclidean space, the sphere, and the Poincarée upper half-plane Audrey Terras

Format:

Book

Author/Creator:

Terras, Audrey, author.

Language:

English

Subjects (All):

Harmonic analysis.

Fourier analysis.

Fourier Analysis.

Medical Subjects:

Fourier Analysis.

Physical Description:

1 online resource

Edition:

Second edition

Place of Publication:

New York Springer 2013

System Details:

PDF

text file

Summary:

This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering. Many corrections, new topics, and updates have been incorporated in this new edition. These include discussions of the work of P. Sarnak and others making progress on various conjectures on modular forms, the work of T. Sunada, Marie-France Vignras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?", Ramanujan graphs, wavelets, quasicrystals, modular knots, triangle and quaternion groups, computations of Maass waveforms, and, finally, the author's comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups, tessellations of H from such discrete group actions, automorphic forms, the Selberg trace formula and its applications in spectral theory as well as number theory

This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering. Many corrections, new topics, and updates have been incorporated in this new edition. These include discussions of the work of P. Sarnak and others making progress on various conjectures on modular forms, the work of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum", Ramanujan graphs, wavelets, quasicrystals, modular knots, triangle and quaternion groups, computations of Maass waveforms, and, finally, the author's comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups Γ, tessellations of H from such discrete group actions, automorphic forms, the Selberg trace formula and its applications in spectral theory as well as number theory

Contents:

Flat Space: Fourier Analysis on R^m A Compact Symmetric Space: The Sphere The Poincaré Upper Half-Plane

Chapter 1 Flat Space. Fourier Analysis on R^m.

1.1 Distributions or Generalized Functions

1.2 Fourier Integrals

1.3 Fourier Series and the Poisson Summation Formula

1.4 Mellin Transforms, Epstein and Dedekind Zeta Functions

1.5 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl's Criterion for Uniform Distribution

Chapter 2 A Compact Symmetric Space

The Sphere

2.1 Fourier Analysis on the Sphere

2.2 O(3) and R^3. The Radon Transform

Chapter 3 The Poincaré Upper Half-Plane

3.1 Hyperbolic Geometry

3.2 Harmonic Analysis on H

3.3 Fundamental Domains for Discrete Subgroups Γ of G = SL(2, R)

3.4 Modular of Automorphic Forms

Classical

3.5 Automorphic Forms

Not So Classical

Maass Waveforms

3.6 Modular Forms and Dirichlet Series. Hecke Theory and Generalizations

References

Index

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed September 16, 2013)

Other Format:

Printed edition:

ISBN:

9781461479727

146147972X

OCLC:

858975843

Access Restriction:

Restricted for use by site license