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The primary goal of these two volumes is to present the theoretical foundation of the field of Euclidean Harmonic analysis. The original edition was published as a single volume, but due to its size, scope, and the addition of new material, the second edition consists of two volumes. The present edition contains a new chapter on time-frequency analysis and the Carleson-Hunt theorem. The first volume contains the classical topics such as Interpolation, Fourier Series, the Fourier Transform, Maximal Functions, Singular Integrals, and Littlewood-Paley Theory. The second volume contains more recent topics such as Function Spaces, Atomic Decompositions, Singular Integrals of Nonconvolution Type, and Weighted Inequalities. These volumes are mainly addressed to graduate students in mathematics and are designed for a two-course sequence on the subject with additional material included for reference. The prerequisites for the first volume are satisfactory completion of courses in real and complex variables. The second volume assumes material from the first. This book is intended to present the selected topics in depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables. About the first edition: "Grafakos's book is very user-friendly with numerous examples illustrating the definitions and ideas... The treatment is thoroughly modern with free use of operators and functional analysis. Morever, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises." - Kenneth Ross, MAA Online

Functional analysis --- Harmonic analysis. Fourier analysis --- Mathematical analysis --- Fourieranalyse --- analyse (wiskunde) --- Fourierreeksen --- functies (wiskunde) --- mathematische modellen --- Harmonic analysis

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The present book is intended as a text for a graduate course on abstract harmonic analysis and its applications. The book can be used as a follow up to Anton Deitmer's previous book, A First Course in Harmonic Analysis, or independently, if the students already have a modest knowledge of Fourier Analysis. In this book, among other things, proofs are given of Pontryagin Duality and the Plancherel Theorem for LCA-groups, which were mentioned but not proved in A First Course in Harmonic Analysis. Using Pontryagin duality, the authors also obtain various structure theorems for locally compact abelian groups. The book then proceeds with Harmonic Analysis on non-abelian groups and its applications to theory in number theory and the theory of wavelets. Knowledge of set theoretic topology, Lebesgue integration, and functional analysis on an introductory level will be required in the body of the book. For the convenience of the reader, all necessary ingredients from these areas have been included in the appendices. Professor Deitmar is Professor of Mathematics at the University of Tübingen, Germany. He is a former Heisenberg fellow and has taught in the U.K. for some years. Professor Echterhoff is Professor of Mathematics and Computer Science at the University of Münster, Germany.

Harmonic analysis. Fourier analysis --- Mathematical analysis --- Fourieranalyse --- analyse (wiskunde) --- Fourierreeksen --- mathematische modellen

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Fondée sur la théorie métaxiale de G. Bonnet et sur l'emploi de la transformation de Fourier fractionnaire, la présentation de l'optique de Fourier adoptée dans cet ouvrage est originale et renouvelle en grande partie le sujet. Outre les thèmes traités habituellement dans ce domaine - diffraction scalaire, formation des images, transfert de la cohérence, holographie, filtrage et corrélation optiques -, le livre inclut une théorie fractionnaire des résonateurs optiques et des faisceaux gaussiens, ou développe encore l'analogie entre diffraction et dispersion de groupe dans les fibres optiques, élargissant de la sorte le champ de la discipline. Issu de l'enseignement dispensé par l'auteur à l'Université et en écoles d'ingénieurs, étayé par ses propres recherches, le livre s'adresse autant à des étudiants en mastère de physique ou des élèves-ingénieurs qu'à des chercheurs ou ingénieurs souhaitant s'initier à l'optique de Fourier fractionnaire. Pierre Pellat-Finet est professeur à l'université de Bretagne-Sud, où il enseigne la physique. Il est également chercheur associé à Télécom Bretagne et professeur associé à l'université nationale de Colombie à Medellin. Ses travaux de recherche portent sur le traitement du signal optique, les télécommunications optiques, la représentation mathématique de la lumière polarisée et l'optique de Fourier fractionnaire.

Harmonic analysis. Fourier analysis --- Optics. Quantum optics --- Physics --- Fourieranalyse --- elektrodynamica --- fysica --- optica

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Functional analysis --- Harmonic analysis. Fourier analysis --- Mathematical analysis --- Fourieranalyse --- analyse (wiskunde) --- Fourierreeksen --- functies (wiskunde) --- mathematische modellen

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Algebra --- Harmonic analysis. Fourier analysis --- Mathematics --- Computer science --- Fourieranalyse --- algebra --- functies (wiskunde) --- informatica --- wiskunde