The Cauchy Transform
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Author |
: Steven R. Bell |
Publisher |
: CRC Press |
Total Pages |
: 221 |
Release |
: 2015-11-04 |
ISBN-10 |
: 9781498727211 |
ISBN-13 |
: 1498727212 |
Rating |
: 4/5 (11 Downloads) |
Synopsis The Cauchy Transform, Potential Theory and Conformal Mapping by : Steven R. Bell
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
Author |
: Xavier Tolsa |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 402 |
Release |
: 2013-12-16 |
ISBN-10 |
: 9783319005966 |
ISBN-13 |
: 3319005960 |
Rating |
: 4/5 (66 Downloads) |
Synopsis Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory by : Xavier Tolsa
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.
Author |
: Joseph A. Cima |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 286 |
Release |
: 2006 |
ISBN-10 |
: 9780821838716 |
ISBN-13 |
: 0821838717 |
Rating |
: 4/5 (16 Downloads) |
Synopsis The Cauchy Transform by : Joseph A. Cima
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.
Author |
: Wolfgang Arendt |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 526 |
Release |
: 2013-11-11 |
ISBN-10 |
: 9783034850759 |
ISBN-13 |
: 3034850751 |
Rating |
: 4/5 (59 Downloads) |
Synopsis Vector-valued Laplace Transforms and Cauchy Problems by : Wolfgang Arendt
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>. .
Author |
: Takafumi Murai |
Publisher |
: Springer |
Total Pages |
: 141 |
Release |
: 2006-11-15 |
ISBN-10 |
: 9783540391050 |
ISBN-13 |
: 3540391053 |
Rating |
: 4/5 (50 Downloads) |
Synopsis A Real Variable Method for the Cauchy Transform, and Analytic Capacity by : Takafumi Murai
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.
Author |
: Xavier Tolsa |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 142 |
Release |
: 2017-01-18 |
ISBN-10 |
: 9781470422523 |
ISBN-13 |
: 1470422522 |
Rating |
: 4/5 (23 Downloads) |
Synopsis Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform by : Xavier Tolsa
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 .
Author |
: Frederick W. King |
Publisher |
: Encyclopedia of Mathematics an |
Total Pages |
: 0 |
Release |
: 2009 |
ISBN-10 |
: 0521517230 |
ISBN-13 |
: 9780521517232 |
Rating |
: 4/5 (30 Downloads) |
Synopsis Hilbert Transforms by : Frederick W. King
The definitive reference on Hilbert transforms covering the mathematical techniques for evaluating them, and their application.
Author |
: Camil Muscalu |
Publisher |
: Cambridge University Press |
Total Pages |
: 341 |
Release |
: 2013-01-31 |
ISBN-10 |
: 9781107031821 |
ISBN-13 |
: 1107031826 |
Rating |
: 4/5 (21 Downloads) |
Synopsis Classical and Multilinear Harmonic Analysis by : Camil Muscalu
This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
Author |
: Takafumi Murai |
Publisher |
: |
Total Pages |
: 152 |
Release |
: 1987 |
ISBN-10 |
: UOM:39015015624110 |
ISBN-13 |
: |
Rating |
: 4/5 (10 Downloads) |
Synopsis A Real Variable Method for the Cauchy Transform and Applications to Analytic Capacity by : Takafumi Murai
Author |
: Sigurdur Helgason |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 214 |
Release |
: 1999-08-01 |
ISBN-10 |
: 0817641092 |
ISBN-13 |
: 9780817641092 |
Rating |
: 4/5 (92 Downloads) |
Synopsis The Radon Transform by : Sigurdur Helgason
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.