Harmonic Analysis On Symmetric Spaces Euclidean Space The Sphere And The Poincaré Upper Half Plane

Author : Audrey Terras
Publisher : Springer Science & Business Media
Page : 413 pages
File Size : 48,7 Mb
Release : 2013-09-12
Category : Mathematics
ISBN : 9781461479727

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Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane by Audrey Terras Pdf

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 and updates have been incorporated in this new edition. Updates include discussions of P. Sarnak and others' work on quantum chaos, 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?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, 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, and the Selberg trace formula and its applications in spectral theory as well as number theory.

Author : Audrey Terras
Publisher : Springer
Page : 500 pages
File Size : 47,5 Mb
Release : 2016-04-26
Category : Mathematics
ISBN : 9781493934089

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

This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups Γ (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant ±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

Author : Audrey Terras
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 44,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461238201

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Harmonic Analysis on Symmetric Spaces and Applications II by Audrey Terras Pdf

Well, finally, here it is-the long-promised "Revenge of the Higher Rank Symmetric Spaces and Their Fundamental Domains." When I began work on it in 1977, I would probably have stopped immediately if someone had told me that ten years would pass before I would declare it "finished." Yes, I am declaring it finished-though certainly not perfected. There is a large amount of work going on at the moment as the piles of preprints reach the ceiling. Nevertheless, it is summer and the ocean calls. So I am not going to spend another ten years revising and polishing. But, gentle reader, do send me your corrections and even your preprints. Thanks to your work, there is an Appendix at the end of this volume with corrections to Volume I. I said it all in the Preface to Volume I. So I will try not to repeat myself here. Yes, the "recent trends" mentioned in that Preface are still just as recent.

Author : Gregory S. Chirikjian,Alexander B. Kyatkin
Publisher : Courier Dover Publications
Page : 881 pages
File Size : 41,9 Mb
Release : 2016-07-20
Category : Mathematics
ISBN : 9780486795645

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Harmonic Analysis for Engineers and Applied Scientists by Gregory S. Chirikjian,Alexander B. Kyatkin Pdf

Although the Fourier transform is among engineering's most widely used mathematical tools, few engineers realize that the extension of harmonic analysis to functions on groups holds great potential for solving problems in robotics, image analysis, mechanics, and other areas. This self-contained approach, geared toward readers with a standard background in engineering mathematics, explores the widest possible range of applications to fields such as robotics, mechanics, tomography, sensor calibration, estimation and control, liquid crystal analysis, and conformational statistics of macromolecules. Harmonic analysis is explored in terms of particular Lie groups, and the text deals with only a limited number of proofs, focusing instead on specific applications and fundamental mathematical results. Forming a bridge between pure mathematics and the challenges of modern engineering, this updated and expanded volume offers a concrete, accessible treatment that places the general theory in the context of specific groups.

Author : Audrey Terras
Publisher : Springer Science & Business Media
Page : 353 pages
File Size : 43,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461251286

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Harmonic Analysis on Symmetric Spaces and Applications I by Audrey Terras Pdf

Since its beginnings with Fourier (and as far back as the Babylonian astron omers), harmonic analysis has been developed with the goal of unraveling the mysteries of the physical world of quasars, brain tumors, and so forth, as well as the mysteries of the nonphysical, but no less concrete, world of prime numbers, diophantine equations, and zeta functions. Quoting Courant and Hilbert, in the preface to the first German edition of Methods of Mathematical Physics: "Recent trends and fashions have, however, weakened the connection between mathematics and physics. " Such trends are still in evidence, harmful though they may be. My main motivation in writing these notes has been a desire to counteract this tendency towards specialization and describe appli cations of harmonic analysis in such diverse areas as number theory (which happens to be my specialty), statistics, medicine, geophysics, and quantum physics. I remember being quite surprised to learn that the subject is useful. My graduate eduation was that of the 1960s. The standard mathematics graduate course proceeded from Definition 1. 1. 1 to Corollary 14. 5. 59, with no room in between for applications, motivation, history, or references to related work. My aim has been to write a set of notes for a very different sort of course.

Author : Valery V. Volchkov,Vitaly V. Volchkov
Publisher : Springer Science & Business Media
Page : 667 pages
File Size : 50,5 Mb
Release : 2009-06-13
Category : Mathematics
ISBN : 9781848825338

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Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group by Valery V. Volchkov,Vitaly V. Volchkov Pdf

The theory of mean periodic functions is a subject which goes back to works of Littlewood, Delsarte, John and that has undergone a vigorous development in recent years. There has been much progress in a number of problems concerning local - pects of spectral analysis and spectral synthesis on homogeneous spaces. The study oftheseproblemsturnsouttobecloselyrelatedtoavarietyofquestionsinharmonic analysis, complex analysis, partial differential equations, integral geometry, appr- imation theory, and other branches of contemporary mathematics. The present book describes recent advances in this direction of research. Symmetric spaces and the Heisenberg group are an active ?eld of investigation at 2 the moment. The simplest examples of symmetric spaces, the classical 2-sphere S 2 and the hyperbolic plane H , play familiar roles in many areas in mathematics. The n Heisenberg groupH is a principal model for nilpotent groups, and results obtained n forH may suggest results that hold more generally for this important class of Lie groups. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces.

Author : Valery V. Volchkov,Vitaly V. Volchkov
Publisher : Springer Science & Business Media
Page : 596 pages
File Size : 40,8 Mb
Release : 2013-01-30
Category : Mathematics
ISBN : 9783034805728

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Offbeat Integral Geometry on Symmetric Spaces by Valery V. Volchkov,Vitaly V. Volchkov Pdf

The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Author : Vladimir I. Bogachev,Oleg G. Smolyanov
Publisher : Springer Nature
Page : 586 pages
File Size : 44,5 Mb
Release : 2020-02-25
Category : Mathematics
ISBN : 9783030382193

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Real and Functional Analysis by Vladimir I. Bogachev,Oleg G. Smolyanov Pdf

This book is based on lectures given at "Mekhmat", the Department of Mechanics and Mathematics at Moscow State University, one of the top mathematical departments worldwide, with a rich tradition of teaching functional analysis. Featuring an advanced course on real and functional analysis, the book presents not only core material traditionally included in university courses of different levels, but also a survey of the most important results of a more subtle nature, which cannot be considered basic but which are useful for applications. Further, it includes several hundred exercises of varying difficulty with tips and references. The book is intended for graduate and PhD students studying real and functional analysis as well as mathematicians and physicists whose research is related to functional analysis.

Author : Steven G. Krantz
Publisher : CRC Press
Page : 547 pages
File Size : 48,8 Mb
Release : 2022-03-07
Category : Mathematics
ISBN : 9781351663069

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Handbook of Complex Analysis by Steven G. Krantz Pdf

In spite of being nearly 500 years old, the subject of complex analysis is still today a vital and active part of mathematics. There are important applications in physics, engineering, and other aspects of technology. This Handbook presents contributed chapters by prominent mathematicians, including the new generation of researchers. More than a compilation of recent results, this book offers students an essential stepping-stone to gain an entry into the research life of complex analysis. Classes and seminars play a role in this process. More, though, is needed for further study. This Handbook will play that role. This book is also a reference and a source of inspiration for more seasoned mathematicians—both specialists in complex analysis and others who want to acquaint themselves with current modes of thought. The chapters in this volume are authored by leading experts and gifted expositors. They are carefully crafted presentations of diverse aspects of the field, formulated for a broad and diverse audience. This volume is a touchstone for current ideas in the broadly construed subject area of complex analysis. It should enrich the literature and point in some new directions.

Author : Tim Hsu
Publisher : American Mathematical Society
Page : 370 pages
File Size : 43,9 Mb
Release : 2023-12-07
Category : Mathematics
ISBN : 9781470476007

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Fourier Series, Fourier Transforms, and Function Spaces by Tim Hsu Pdf

Fourier Series, Fourier Transforms, and Function Spaces is designed as a textbook for a second course or capstone course in analysis for advanced undergraduate or beginning graduate students. By assuming the existence and properties of the Lebesgue integral, this book makes it possible for students who have previously taken only one course in real analysis to learn Fourier analysis in terms of Hilbert spaces, allowing for both a deeper and more elegant approach. This approach also allows junior and senior undergraduates to study topics like PDEs, quantum mechanics, and signal processing in a rigorous manner. Students interested in statistics (time series), machine learning (kernel methods), mathematical physics (quantum mechanics), or electrical engineering (signal processing) will find this book useful. With 400 problems, many of which guide readers in developing key theoretical concepts themselves, this text can also be adapted to self-study or an inquiry-based approach. Finally, of course, this text can also serve as motivation and preparation for students going on to further study in analysis.

Author : Šárka Hošková-Mayerová,Cristina Flaut,Fabrizio Maturo
Publisher : Springer Nature
Page : 515 pages
File Size : 42,5 Mb
Release : 2021-01-13
Category : Technology & Engineering
ISBN : 9783030613341

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Algorithms as a Basis of Modern Applied Mathematics by Šárka Hošková-Mayerová,Cristina Flaut,Fabrizio Maturo Pdf

This book offers a self-contained guide to advanced algorithms and their applications in various fields of science. Gathering contributions by authoritative researchers in the field of mathematics, statistics and computer science, it aims at offering a comprehensive and up-to-date view of algorithms, including the theory behind them, as well as practical considerations, current limitations and solutions. It covers applications in energy management, decision making, computer networks, materials science, mechanics and process optimization. It offers an integrated and timely guide to important algorithms, and represents a valuable reference resource for graduate students and researchers in various fields of applied mathematics, statistics and engineering.

Author : Audrey Terras
Publisher : Cambridge University Press
Page : 331 pages
File Size : 40,6 Mb
Release : 2018-12-20
Category : Mathematics
ISBN : 9781107164079

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Abstract Algebra with Applications by Audrey Terras Pdf

This text offers a friendly and concise introduction to abstract algebra, emphasizing its uses in the modern world.

Author : Henrik Schlichtkrull
Publisher : Springer Science & Business Media
Page : 197 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461252986

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Hyperfunctions and Harmonic Analysis on Symmetric Spaces by Henrik Schlichtkrull Pdf

This book gives an introductory exposition of the theory of hyperfunctions and regular singularities. This first English introduction to hyperfunctions brings readers to the forefront of research in the theory of harmonic analysis on symmetric spaces. A substantial bibliography is also included. This volume is based on a paper which was awarded the 1983 University of Copenhagen Gold Medal Prize.

Author : Audrey Terras
Publisher : Springer
Page : 128 pages
File Size : 50,5 Mb
Release : 1985-07-01
Category : Mathematics
ISBN : 0387961593

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Harmonic Analysis on Symmetric Spaces and Applications I by Audrey Terras Pdf

Author : Guido Weiss,Stephen Wainger,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 460 pages
File Size : 44,7 Mb
Release : 1979
Category : Mathematics
ISBN : 9780821814369

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Harmonic Analysis in Euclidean Spaces by Guido Weiss,Stephen Wainger,American Mathematical Society Pdf

Contains sections on Real harmonic analysis, Hardy spaces and BMO,Harmonic functions, potential theory and theory of functions of one complex variable