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The book is written for graduate students who have read the first book and like to see the proofs which were not given there and/or want to see the full extent of the theory. On the other hand it can be read independently from the first one, only a modest knowledge on Fourier series/tranform is required to understand the examples.

This book fills a major gap in the textbook literature, as a full proof of Pontryagin Duality and Plancherel Theorem is hard to come by. It is usually given in books that focus on C*-algebras and thus carry too much technical overload for the reader who only wants these basic results of Harmonic Analysis. Other proofs use the structure theory which carries the reader away in a different direction. Here the authors consider the Banach-algebra approach more elegant and enlighting. They provide a streamlined approach that reaches the main results directly, and they also give the generalizations to the non-Abelian case.

Another main pillar of Harmonic analysis is the Poisson Summation Formula. We give its generalization to LCA-groups. The Selberg Trace Formula is considered the generalization of the Poisson Formula to non-abelian groups. The authors give the first textbook approach to this deep and useful formula in full generality. The last two chapters are devoted to examples of applications of the Selberg Trace Formula.

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This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and their use in both applied and classical harmonic analysis. The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory.Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses. The self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications. ... Read more

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Sampling, wavelets, and tomography are three active areas of contemporary mathematics sharing common roots that lie at the heart of harmonic and Fourier analysis.The advent of new techniques in mathematical analysis has strengthened their interdependence and led to some new and interesting results in the field.

This state-of-the-art book not only presents new results in these research areas, but it also demonstrates the role of sampling in both wavelet theory and tomography.Specific topics covered include:

* Robustness of Regular Sampling in Sobolev Algebras* Irregular and Semi-Irregular Weyl-Heisenberg Frames* Adaptive Irregular Sampling in Meshfree Flow Simulation * Sampling Theorems for Non-Bandlimited Signals * Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets* Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces* Sampling Theory and Parallel-Beam Tomography * Thin-Plate Spline Interpolation in Medical Imaging* Filtered Back-Projection Algorithms for Spiral Cone Computed Tomography

Aimed at mathematicians, scientists, and engineers working in signal and image processing and medical imaging, the work is designed to be accessible to an audience with diverse mathematical backgrounds.Although the volume reflects the contributions of renowned mathematicians and engineers, each chapter has an expository introduction written for the non-specialist.One of the key features of the book is an introductory chapter stressing the interdependence of the three main areas covered.A comprehensive index completes the work.

Contributors:

J.J. BenedettoN.K. BoseP.G. CasazzaY.C. EldarH.G. FeichtingerA. FaridaniA. IskeS. JaffardA. KatsevichS. LertrattanapanichG. LauritschB. MairM. PapadakisP.P. VaidyanathanT. WertherD.C. WilsonA.I. Zayed ... Read more

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This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis.

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This volume is devoted to the theme of Noncommutative Harmonic Analysis and consists of articles in honor of Jacques Carmona, whose scientific interests range through all aspects of Lie group representations. Thus, the topics encompass the theory of representations of reductive Lie groups, and especially the determination of the unitary dual, the problem of geometric realizations of representations, harmonic analysis on reductive symmetric spaces, the study of automorphic forms, and results in harmonic analysis that apply to the Langlands program.

General Lie groups is also discussed, particularly from the orbit method perspective, which has been a constant source of inspiration for both the theory of reductive Lie groups and for general Lie groups. Also covered is Kontsevich quantization, which has appeared in the recent years as a powerful tool.Contributors to this volume include: Baldoni, Barbasch, Barchini, Bieliavsky, Bouaziz, Delorme, Van Den Ban, Harinck, Hersant, Khalgui, Knapp, Kostant, Libine, Mantini, Pevzner, Rossmann, Rubenthaler, Schmid, Torasso, Vergne. ... Read more

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This book brings together five papers that have been influential in the study of Lie groups. Though published more than 20 years ago, these papers made fundamental contributions that deserve much broader exposure. In addition, the subsequent literature that has subsumed these papers cannot replace the originality and vitality they contain. The editors have provided a brief introduction to each paper, as well as a synopsis of the major developments which have occurred in the area covered by each paper. Included here are the doctoral theses of Arthur, Osborne, and Schmid. Arthur's thesis is closely related to Trombi's paper insofar as both deal with harmonic analysis on real semisimple Lie groups, and, in particular, analysis on the Schwartz space of Harish-Chandra. Arthur's thesis is concerned with the image under the Fourier transform of the Schwartz space of a semisimple Lie group of real rank one, while Trombi's paper provides an expository account of the harmonic analysis associated to the decomposition of the Schwartz space under the regular representation. In his thesis, Osborne extends the Atiyah-Bott fixed point theorem for elliptic complexes to obtain a fixed point formula for complexes that are not elliptic. Schmid proves a generalization of the Borel-Weil theorem concerning an explicit and geometric realization of the irreducible representations of a compact, connected semisimple Lie group. Langlands's fundamental paper provides a classification of irreducible, admissible representations of real reductive Lie groups. ... Read more

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Harish-Chandra was a mathematician of great power, vision, and remarkable ingenuity. His profound contributions to the representation theory of Lie groups, harmonic analysis, and related areas left researchers a rich legacy that continues today. This book presents the proceedings of an AMS Special Session entitled, "Representation Theory and Noncommutative Harmonic Analysis: A Special Session Honoring the Memory of Harish-Chandra", which marked 75 years since his birth and 15 years since his untimely death at age 60.

Contributions to the volume were written by an outstanding group of internationally known mathematicians. Included are expository and historical surveys and original research papers. The book also includes talks given at the IAS Memorial Service in 1983 by colleagues who knew Harish-Chandra well. Also reprinted are two articles entitled, "Some Recollections of Harish-Chandra", by A. Borel, and "Harish-Chandra's c-Function: A Mathematical Jewel", by S. Helgason. In addition, an expository paper, "An Elementary Introduction to Harish-Chandra's Work", gives an overview of some of his most basic mathematical ideas with references for further study.

This volume offers a comprehensive retrospective of Harish-Chandra's professional life and work. Personal recollections give the book particular significance. Readers should have an advanced-level background in the representation theory of Lie groups and harmonic analysis. ... Read more

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This two-part survey provides a short review of the classical part of representation theory, carefully exposing the structure of the theory without overwhelming readers with details, and deals with representations of Virasoro and Kac-Moody algebra. It presents a wealth of recent results on representations of infinite-dimensional groups.

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The present book is a continuation of the three-volume workRepresentation of Lie Groups and Special Functions by the sameauthors. Here, they deal with the exposition of the main newdevelopments in the contemporary theory of multivariate specialfunctions, bringing together material that has not been presented inmonograph form before. The theory of orthogonal symmetric polynomials (Jack polynomials,Macdonald's polynomials and others) and multivariate hypergeometricfunctions associated to symmetric polynomials are treated.Multivariate hypergeometric functions, multivariate Jacobi polynomialsand h-harmonic polynomials connected with root systems andCoxeter groups are introduced. Also, the theory of Gel'fandhypergeometric functions and the theory of multivariate hypergeometricseries associated to Clebsch--Gordan coefficients of the unitary groupU(n) is given. The volume concludes with an extensivebibliography. For research mathematicians and physicists, postgraduate students inmathematics and mathematical and theoretical physics. ... Read more

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This book offers a concise introduction to the rapidlyexpanding field of infinite dimensional stochastic analysis. It treatsMalliavin calculus and white noise analysis in a single book,presenting these two different areas in a unified setting of Gaussianprobability spaces. Topics include recent results and developments inthe areas of quasi-sure analysis, anticipating stochastic calculus,generalised operator theory and applications in quantum physics. Ashort overview on the foundations of infinite dimensional analysis isgiven. Audience: This volume will be of interest to researchers andgraduate students whose work involves probability theory, stochasticprocesses, functional analysis, operator theory, mathematics ofphysics and abstract harmonic analysis. ... Read more

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The main goal of this book is to demonstrate the usefulness ofset-theoretical methods in various questions of real analysis andclassical measure theory. In this context, many statements and factsfrom analysis are treated as consequences of purely set-theoreticalassertions which can successfully be applied to measures and Bairecategory. Topics covered include similarities and differences betweenmeasure and category; constructions of nonmeasurable sets and of setswithout the Baire property; three aspects of the measure extensionproblem; the principle of condensation of singularities from the pointof view of the Kuratowski-Ulam theorem; transformation groupsand invariant (quasi-invariant) measures; the uniqueness property ofan invariant measure; and ordinary differential equations withnonmeasurable right-hand sides.
Audience: The material presented in the book is essentiallyself-contained and is accessible to a wide audience of mathematicians.It will appeal to specialists in set theory, mathematical analysis,measure theory and general topology. It can also be recommended as atextbook for postgraduate students who are interested in theapplications of set-theoretical methods to the above-mentioned domainsof mathematics. ... Read more