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The Cauchy Transform
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Book Synopsis The Cauchy Transform, Potential Theory and Conformal Mapping by : Steven R. Bell
Download or read book The Cauchy Transform, Potential Theory and Conformal Mapping written by Steven R. Bell and published by CRC Press. This book was released on 2015-11-04 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Cauchy Transform, Potential Theory and Conformal Mapping explores the most central result in all of classical function theory, the Cauchy integral formula, in a new and novel way based on an advance made by Kerzman and Stein in 1976.The book provides a fast track to understanding the Riemann Mapping Theorem. The Dirichlet and Neumann problems f
Book Synopsis The Cauchy Transform by : Joseph A. Cima
Download or read book The Cauchy Transform written by Joseph A. Cima and published by American Mathematical Soc.. This book was released on 2006 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures.
Book Synopsis Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory by : Xavier Tolsa
Download or read book Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory written by Xavier Tolsa and published by Springer Science & Business Media. This book was released on 2013-12-16 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995–2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderón-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painlevé problem, which was first posed around 1900, consists in finding a description of the removable singularities for bounded analytic functions in metric and geometric terms. Analytic capacity is a key tool in the study of this problem. In the 1960s Vitushkin conjectured that the removable sets which have finite length coincide with those which are purely unrectifiable. Moreover, because of the applications to the theory of uniform rational approximation, he posed the question as to whether analytic capacity is semiadditive. This work presents full proofs of Vitushkin’s conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the Cauchy transforms and other singular integrals. The book is largely self-contained and should be accessible for graduate students in analysis, as well as a valuable resource for researchers.
Book Synopsis The Cauchy Transform, Potential Theory and Conformal Mapping by : Steven R. Bell
Download or read book The Cauchy Transform, Potential Theory and Conformal Mapping written by Steven R. Bell and published by CRC Press. This book was released on 1992-08-14 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Cauchy integral formula is the most central result in all of classical function theory. A recent discovery of Kerzman and Stein allows more theorems than ever to be deduced from simple facts about the Cauchy integral. In this book, the Riemann Mapping Theorem is deduced, the Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernal is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely using formulas stemming from the Kerzman-Stein result. These explicit formulas yield new numerical methods for computing the classical objects of potential theory and conformal mapping, and the book provides succinct, complete explanations of these methods. The Cauchy Transform, Potential Theory, and Conformal Mapping is suitable for pure and applied math students taking a beginning graduate-level topics course on aspects of complex analysis. It will also be useful to physicists and engineers interested in a clear exposition on a fundamental topic of complex analysis, methods, and their application.
Book Synopsis Vector-valued Laplace Transforms and Cauchy Problems by : Wolfgang Arendt
Download or read book Vector-valued Laplace Transforms and Cauchy Problems written by Wolfgang Arendt and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: Linear evolution equations in Banach spaces have seen important developments in the last two decades. This is due to the many different applications in the theory of partial differential equations, probability theory, mathematical physics, and other areas, and also to the development of new techniques. One important technique is given by the Laplace transform. It played an important role in the early development of semigroup theory, as can be seen in the pioneering monograph by Rille and Phillips [HP57]. But many new results and concepts have come from Laplace transform techniques in the last 15 years. In contrast to the classical theory, one particular feature of this method is that functions with values in a Banach space have to be considered. The aim of this book is to present the theory of linear evolution equations in a systematic way by using the methods of vector-valued Laplace transforms. It is simple to describe the basic idea relating these two subjects. Let A be a closed linear operator on a Banach space X. The Cauchy problern defined by A is the initial value problern (t 2 0), (CP) {u'(t) = Au(t) u(O) = x, where x E X is a given initial value. If u is an exponentially bounded, continuous function, then we may consider the Laplace transform 00 u(>. ) = 1 e-). . tu(t) dt of u for large real>. .
Book Synopsis Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform by : Xavier Tolsa
Download or read book Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform written by Xavier Tolsa and published by American Mathematical Soc.. This book was released on 2017-01-18 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the proof of two related results. The first one asserts that if is a Radon measure in satisfyingfor -a.e. , then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only ifH^1x2EThe second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square .
Book Synopsis A Real Variable Method for the Cauchy Transform, and Analytic Capacity by : Takafumi Murai
Download or read book A Real Variable Method for the Cauchy Transform, and Analytic Capacity written by Takafumi Murai and published by Springer. This book was released on 2006-11-15 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research monograph studies the Cauchy transform on curves with the object of formulating a precise estimate of analytic capacity. The note is divided into three chapters. The first chapter is a review of the Calderón commutator. In the second chapter, a real variable method for the Cauchy transform is given using only the rising sun lemma. The final and principal chapter uses the method of the second chapter to compare analytic capacity with integral-geometric quantities. The prerequisites for reading this book are basic knowledge of singular integrals and function theory. It addresses specialists and graduate students in function theory and in fluid dynamics.
Book Synopsis Hilbert Transforms by : Frederick W. King
Download or read book Hilbert Transforms written by Frederick W. King and published by Encyclopedia of Mathematics an. This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The definitive reference on Hilbert transforms covering the mathematical techniques for evaluating them, and their application.
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 The Radon Transform by : Sigurdur Helgason
Download or read book The Radon Transform written by Sigurdur Helgason and published by Springer Science & Business Media. This book was released on 1999-08-01 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representations, X-ray technology, nuclear magnetic resonance scanning, and tomography. This second edition, significantly expanded and updated, presents new material taking into account some of the progress made in the field since 1980. Aimed at beginning graduate students, this monograph will be useful in the classroom or as a resource for self-study. Readers will find here an accessible introduction to Radon transform theory, an elegant topic in integral geometry.
Book Synopsis The Hilbert Transform of Schwartz Distributions and Applications by : J. N. Pandey
Download or read book The Hilbert Transform of Schwartz Distributions and Applications written by J. N. Pandey and published by John Wiley & Sons. This book was released on 2011-10-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a modern and up-to-date treatment of the Hilberttransform of distributions and the space of periodic distributions.Taking a simple and effective approach to a complex subject, thisvolume is a first-rate textbook at the graduate level as well as anextremely useful reference for mathematicians, applied scientists,and engineers. The author, a leading authority in the field, shares with thereader many new results from his exhaustive research on the Hilberttransform of Schwartz distributions. He describes in detail how touse the Hilbert transform to solve theoretical and physicalproblems in a wide range of disciplines; these include aerofoilproblems, dispersion relations, high-energy physics, potentialtheory problems, and others. Innovative at every step, J. N. Pandey provides a new definitionfor the Hilbert transform of periodic functions, which isespecially useful for those working in the area of signalprocessing for computational purposes. This definition could alsoform the basis for a unified theory of the Hilbert transform ofperiodic, as well as nonperiodic, functions. The Hilbert transform and the approximate Hilbert transform ofperiodic functions are worked out in detail for the first time inbook form and can be used to solve Laplace's equation with periodicboundary conditions. Among the many theoretical results proved inthis book is a Paley-Wiener type theorem giving thecharacterization of functions and generalized functions whoseFourier transforms are supported in certain orthants of Rn. Placing a strong emphasis on easy application of theory andtechniques, the book generalizes the Hilbert problem in higherdimensions and solves it in function spaces as well as ingeneralized function spaces. It simplifies the one-dimensionaltransform of distributions; provides solutions to thedistributional Hilbert problems and singular integral equations;and covers the intrinsic definition of the testing function spacesand its topology. The book includes exercises and review material for all majortopics, and incorporates classical and distributional problems intothe main text. Thorough and accessible, it explores new ways to usethis important integral transform, and reinforces its value in bothmathematical research and applied science. The Hilbert transform made accessible with many new formulas anddefinitions Written by today's foremost expert on the Hilbert transform ofgeneralized functions, this combined text and reference covers theHilbert transform of distributions and the space of periodicdistributions. The author provides a consistently accessibletreatment of this advanced-level subject and teaches techniquesthat can be easily applied to theoretical and physical problemsencountered by mathematicians, applied scientists, and graduatestudents in mathematics and engineering. Introducing many new inversion formulas that have been developedand applied by the author and his research associates, the book: * Provides solutions to the distributional Hilbert problem andsingular integral equations * Focuses on the Hilbert transform of Schwartz distributions,giving intrinsic definitions of the space H(D) and its topology * Covers the Paley-Wiener theorem and provides many importanttheoretical results of importance to research mathematicians * Provides the characterization of functions and generalizedfunctions whose Fourier transforms are supported in certainorthants of Rn * Offers a new definition of the Hilbert transform of the periodicfunction that can be used for computational purposes in signalprocessing * Develops the theory of the Hilbert transform of periodicdistributions and the approximate Hilbert transform of periodicdistributions * Provides exercises at the end of each chapter--useful toprofessors in planning assignments, tests, and problems
Book Synopsis A Real Variable Method for the Cauchy Transform and Applications to Analytic Capacity by : Takafumi Murai
Download or read book A Real Variable Method for the Cauchy Transform and Applications to Analytic Capacity written by Takafumi Murai and published by . This book was released on 1987 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Hilbert Transform in Fourier Spectroscopy by : Hajime Sakai
Download or read book Hilbert Transform in Fourier Spectroscopy written by Hajime Sakai and published by . This book was released on 1966 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Singular Integrals and Differentiability Properties of Functions (PMS-30), Volume 30 by : Elias M. Stein
Download or read book Singular Integrals and Differentiability Properties of Functions (PMS-30), Volume 30 written by Elias M. Stein and published by Princeton University Press. This book was released on 2016-06-02 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.
Book Synopsis Hypercomplex Analysis by : Irene Sabadini
Download or read book Hypercomplex Analysis written by Irene Sabadini and published by Springer Science & Business Media. This book was released on 2009-04-21 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains selected papers from the ISAAC conference 2007 and invited contributions. This book covers various topics that represent the main streams of research in hypercomplex analysis as well as the expository articles. It is suitable for researchers and postgraduate students in various areas of mathematical analysis.
Book Synopsis Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions by : Thomas Trogdon
Download or read book Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions written by Thomas Trogdon and published by SIAM. This book was released on 2015-12-22 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Riemann?Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann?Hilbert problem.This book, the most comprehensive one to date on the applied and computational theory of Riemann?Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann?Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann?Hilbert method, each of mathematical or physical significance or both.?
Book Synopsis The Cauchy Problem in General Relativity by : Hans Ringström
Download or read book The Cauchy Problem in General Relativity written by Hans Ringström and published by European Mathematical Society. This book was released on 2009 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein's equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann-Lemaitre-Robertson-Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein's equations as an initial value problem allows a closer study of their solutions. This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those without prior background in the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.