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Classical Fourier Analysis
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Book Synopsis Classical Fourier Analysis by : Loukas Grafakos
Download or read book Classical Fourier Analysis written by Loukas Grafakos and published by Springer Science & Business Media. This book was released on 2008-09-18 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some 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. While the 1st edition was published as a single volume, the new edition will contain 120 pp of new material, with an additional chapter on time-frequency analysis and other modern topics. As a result, the book is now being published in 2 separate volumes, the first volume containing the classical topics (Lp Spaces, Littlewood-Paley Theory, Smoothness, etc...), the second volume containing the modern topics (weighted inequalities, wavelets, atomic decomposition, etc...). From a review of the first edition: “Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. 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.” - Ken Ross, MAA Online
Book Synopsis Classical Fourier Analysis by : Loukas Grafakos
Download or read book Classical Fourier Analysis written by Loukas Grafakos and published by Springer. This book was released on 2014-11-17 with total page 647 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goal of this text is to present the theoretical foundation of the field of Fourier analysis on Euclidean spaces. It covers classical topics such as interpolation, Fourier series, the Fourier transform, maximal functions, singular integrals, and Littlewood–Paley theory. The primary readership is intended to be graduate students in mathematics with the prerequisite including satisfactory completion of courses in real and complex variables. The coverage of topics and exposition style are designed to leave no gaps in understanding and stimulate further study. This third edition includes new Sections 3.5, 4.4, 4.5 as well as a new chapter on “Weighted Inequalities,” which has been moved from GTM 250, 2nd Edition. Appendices I and B.9 are also new to this edition. Countless corrections and improvements have been made to the material from the second edition. Additions and improvements include: more examples and applications, new and more relevant hints for the existing exercises, new exercises, and improved references.
Book Synopsis Fourier Analysis by : Javier Duoandikoetxea Zuazo
Download or read book Fourier Analysis written by Javier Duoandikoetxea Zuazo and published by American Mathematical Soc.. This book was released on 2001-01-01 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fourier analysis encompasses a variety of perspectives and techniques. This volume presents the real variable methods of Fourier analysis introduced by Calderón and Zygmund. The text was born from a graduate course taught at the Universidad Autonoma de Madrid and incorporates lecture notes from a course taught by José Luis Rubio de Francia at the same university. Motivated by the study of Fourier series and integrals, classical topics are introduced, such as the Hardy-Littlewood maximal function and the Hilbert transform. The remaining portions of the text are devoted to the study of singular integral operators and multipliers. Both classical aspects of the theory and more recent developments, such as weighted inequalities, H1, BMO spaces, and the T1 theorem, are discussed. Chapter 1 presents a review of Fourier series and integrals; Chapters 2 and 3 introduce two operators that are basic to the field: the Hardy-Littlewood maximal function and the Hilbert transform in higher dimensions. Chapters 4 and 5 discuss singular integrals, including modern generalizations. Chapter 6 studies the relationship between H1, BMO, and singular integrals; Chapter 7 presents the elementary theory of weighted norm inequalities. Chapter 8 discusses Littlewood-Paley theory, which had developments that resulted in a number of applications. The final chapter concludes with an important result, the T1 theorem, which has been of crucial importance in the field. This volume has been updated and translated from the original Spanish edition (1995). Minor changes have been made to the core of the book; however, the sections, "Notes and Further Results" have been considerably expanded and incorporate new topics, results, and references. It is geared toward graduate students seeking a concise introduction to the main aspects of the classical theory of singular operators and multipliers. Prerequisites include basic knowledge in Lebesgue integrals and functional analysis.
Book Synopsis Classical and Multilinear Harmonic Analysis by : Camil Muscalu
Download or read book Classical and Multilinear Harmonic Analysis written by Camil Muscalu and published by Cambridge University Press. This book was released on 2013-01-31 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
Book Synopsis Fourier Integrals in Classical Analysis by : Christopher Donald Sogge
Download or read book Fourier Integrals in Classical Analysis written by Christopher Donald Sogge and published by Cambridge University Press. This book was released on 1993-02-26 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: An advanced monograph concerned with modern treatments of central problems in harmonic analysis.
Book Synopsis Fourier Analysis and Approximation of Functions by : Roald M. Trigub
Download or read book Fourier Analysis and Approximation of Functions written by Roald M. Trigub and published by Springer Science & Business Media. This book was released on 2004-09-07 with total page 610 pages. Available in PDF, EPUB and Kindle. Book excerpt: In Fourier Analysis and Approximation of Functions basics of classical Fourier Analysis are given as well as those of approximation by polynomials, splines and entire functions of exponential type. In Chapter 1 which has an introductory nature, theorems on convergence, in that or another sense, of integral operators are given. In Chapter 2 basic properties of simple and multiple Fourier series are discussed, while in Chapter 3 those of Fourier integrals are studied. The first three chapters as well as partially Chapter 4 and classical Wiener, Bochner, Bernstein, Khintchin, and Beurling theorems in Chapter 6 might be interesting and available to all familiar with fundamentals of integration theory and elements of Complex Analysis and Operator Theory. Applied mathematicians interested in harmonic analysis and/or numerical methods based on ideas of Approximation Theory are among them. In Chapters 6-11 very recent results are sometimes given in certain directions. Many of these results have never appeared as a book or certain consistent part of a book and can be found only in periodics; looking for them in numerous journals might be quite onerous, thus this book may work as a reference source. The methods used in the book are those of classical analysis, Fourier Analysis in finite-dimensional Euclidean space Diophantine Analysis, and random choice.
Book Synopsis Classical and Modern Fourier Analysis by : Loukas Grafakos
Download or read book Classical and Modern Fourier Analysis written by Loukas Grafakos and published by Prentice Hall. This book was released on 2004 with total page 968 pages. Available in PDF, EPUB and Kindle. Book excerpt: An ideal refresher or introduction to contemporary Fourier Analysis, this book starts from the beginning and assumes no specific background. Readers gain a solid foundation in basic concepts and rigorous mathematics through detailed, user-friendly explanations and worked-out examples, acquire deeper understanding by working through a variety of exercises, and broaden their applied perspective by reading about recent developments and advances in the subject. Features over 550 exercises with hints (ranging from simple calculations to challenging problems), illustrations, and a detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals ofL p functions --one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae.L p Spaces and Interpolation. Maximal Functions, Fourier transforms, and Distributions. Fourier Analysis on the Torus. Singular Integrals of Convolution Type. Littlewood-Paley Theory and Multipliers. Smoothness and Function Spaces.BMO and Carleson Measures. Singular Integrals of Nonconvolution Type. Weighted Inequalities. Boundedness and Convergence of Fourier Integrals. For mathematicians interested in harmonic analysis.
Book Synopsis Numerical Fourier Analysis by : Gerlind Plonka
Download or read book Numerical Fourier Analysis written by Gerlind Plonka and published by Springer. This book was released on 2019-02-05 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a unified presentation of Fourier theory and corresponding algorithms emerging from new developments in function approximation using Fourier methods. It starts with a detailed discussion of classical Fourier theory to enable readers to grasp the construction and analysis of advanced fast Fourier algorithms introduced in the second part, such as nonequispaced and sparse FFTs in higher dimensions. Lastly, it contains a selection of numerical applications, including recent research results on nonlinear function approximation by exponential sums. The code of most of the presented algorithms is available in the authors’ public domain software packages. Students and researchers alike benefit from this unified presentation of Fourier theory and corresponding algorithms.
Book Synopsis Fourier Analysis by : Elias M. Stein
Download or read book Fourier Analysis written by Elias M. Stein and published by Princeton University Press. This book was released on 2011-02-11 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
Book Synopsis Principles of Fourier Analysis by : Kenneth B. Howell
Download or read book Principles of Fourier Analysis written by Kenneth B. Howell and published by CRC Press. This book was released on 2016-12-12 with total page 742 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas. Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented. Principles of Fourier Analysis stimulates the appreciation and understanding of the fundamental concepts and serves both beginning students who have seen little or no Fourier analysis as well as the more advanced students who need a deeper understanding. Insightful, non-rigorous derivations motivate much of the material, and thought-provoking examples illustrate what can go wrong when formulas are misused. With clear, engaging exposition, readers develop the ability to intelligently handle the more sophisticated mathematics that Fourier analysis ultimately requires.
Book Synopsis Fourier Analysis in Probability Theory by : Tatsuo Kawata
Download or read book Fourier Analysis in Probability Theory written by Tatsuo Kawata and published by Academic Press. This book was released on 2014-06-17 with total page 681 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fourier Analysis in Probability Theory provides useful results from the theories of Fourier series, Fourier transforms, Laplace transforms, and other related studies. This 14-chapter work highlights the clarification of the interactions and analogies among these theories. Chapters 1 to 8 present the elements of classical Fourier analysis, in the context of their applications to probability theory. Chapters 9 to 14 are devoted to basic results from the theory of characteristic functions of probability distributors, the convergence of distribution functions in terms of characteristic functions, and series of independent random variables. This book will be of value to mathematicians, engineers, teachers, and students.
Book Synopsis Classical and Multilinear Harmonic Analysis by : Camil Muscalu
Download or read book Classical and Multilinear Harmonic Analysis written by Camil Muscalu and published by Cambridge University Press. This book was released on 2013-01-31 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
Book Synopsis Early Fourier Analysis by : Hugh L. Montgomery
Download or read book Early Fourier Analysis written by Hugh L. Montgomery and published by American Mathematical Soc.. This book was released on 2014-12-10 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fourier Analysis is an important area of mathematics, especially in light of its importance in physics, chemistry, and engineering. Yet it seems that this subject is rarely offered to undergraduates. This book introduces Fourier Analysis in its three most classical settings: The Discrete Fourier Transform for periodic sequences, Fourier Series for periodic functions, and the Fourier Transform for functions on the real line. The presentation is accessible for students with just three or four terms of calculus, but the book is also intended to be suitable for a junior-senior course, for a capstone undergraduate course, or for beginning graduate students. Material needed from real analysis is quoted without proof, and issues of Lebesgue measure theory are treated rather informally. Included are a number of applications of Fourier Series, and Fourier Analysis in higher dimensions is briefly sketched. A student may eventually want to move on to Fourier Analysis discussed in a more advanced way, either by way of more general orthogonal systems, or in the language of Banach spaces, or of locally compact commutative groups, but the experience of the classical setting provides a mental image of what is going on in an abstract setting.
Book Synopsis Harmonic Analysis by : María Cristina Pereyra
Download or read book Harmonic Analysis written by María Cristina Pereyra and published by American Mathematical Soc.. This book was released on 2012 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conveys the remarkable beauty and applicability of the ideas that have grown from Fourier theory. It presents for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization).
Book Synopsis Complex Analysis by : Elias M. Stein
Download or read book Complex Analysis written by Elias M. Stein and published by Princeton University Press. This book was released on 2010-04-22 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
Book Synopsis Fourier Analysis on Groups by : Walter Rudin
Download or read book Fourier Analysis on Groups written by Walter Rudin and published by Courier Dover Publications. This book was released on 2017-04-19 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written by a master mathematical expositor, this classic text reflects the results of the intense period of research and development in the area of Fourier analysis in the decade preceding its first publication in 1962. The enduringly relevant treatment is geared toward advanced undergraduate and graduate students and has served as a fundamental resource for more than five decades. The self-contained text opens with an overview of the basic theorems of Fourier analysis and the structure of locally compact Abelian groups. Subsequent chapters explore idempotent measures, homomorphisms of group algebras, measures and Fourier transforms on thin sets, functions of Fourier transforms, closed ideals in L1(G), Fourier analysis on ordered groups, and closed subalgebras of L1(G). Helpful Appendixes contain background information on topology and topological groups, Banach spaces and algebras, and measure theory.
Book Synopsis Fourier Analysis and Its Applications by : Anders Vretblad
Download or read book Fourier Analysis and Its Applications written by Anders Vretblad and published by Springer Science & Business Media. This book was released on 2006-04-18 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: A carefully prepared account of the basic ideas in Fourier analysis and its applications to the study of partial differential equations. The author succeeds to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral. Readers should be familiar with calculus, linear algebra, and complex numbers. At the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level. A variety of worked examples and exercises will help the readers to apply their newly acquired knowledge.
