On the Boundary Behavior of Holomorphic Functions in the Unit Disk PDF Download
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Author: Sister Claude Marie Faust
Publisher:
ISBN:
Category :
Languages : en
Pages : 112
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Author: Sister Claude Marie Faust
Publisher:
ISBN:
Category :
Languages : en
Pages : 112
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Author: Elias M. Stein
Publisher: Princeton University Press
ISBN: 1400871263
Category : Mathematics
Languages : en
Pages : 83
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Book Description
This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field. Originally published in 1972. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
![Boundary Behavior of Holomorphic Functions in the Unit Ball of C[subscript N] PDF](https://books.google.com/books/content/images/frontcover/9iPcNwAACAAJ?fife=w80-h100)
Author: Paula A. Russo
Publisher:
ISBN:
Category : Holomorphic functions
Languages : en
Pages : 170
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Author: Fausto Di Biase
Publisher: Birkhauser
ISBN: 9780817642990
Category : Mathematics
Languages : en
Pages : 300
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Book Description
This monograph examines the boundary behavior of holomorphic functions in several complex variables. Moving beyond the early ideas of Fatou and others, Koranyi and then Stein in the late 1960s and early 1970s deepened the study of Fatou-type theorems in several complex variables, showing that in a general context, approach regions of a shape dramatically larger than non-tangential will give rise to a Fatou-type theorem. These have become known as the admissible regions of Koranyi and Stein. It turns out, however, that the admissible approach regions are only optimal on strongly pseudoconvex domains. Considerable effort has been made in the last 20 years to adapt Fatou theory, and the approach regions in particular, to the Levi geometry of a given domain in multidimensional complex space. The work of Di Biase in the late 1990s is devoted to the Nagel--Stein phenomenon, describing a more general notion of approach region that supersedes the classical ideas of non-tangential and admissible. Krantz's work Function Theory of Several Complex Variables (2000), still the only introduction to the subject, focuses on methods based on maximal function estimates. To date, the main open problem, which is the special focus of this book, is the issue of determining the {it optimal natural approach regions} for the almost everywhere convergence to the boundary of certain smoothly bounded pseudoconvex domains. This book provides the proper framework for the eventual solution of the main problem. This work gives an updated, comprehensive, and self-contained exposition of many results that have never appeared in book form. Starting with foundational material, i.e., from the unit disc in one complexvariable, the reader is lead to the latest discoveries in higher dimensions. New results in boundary value issues of holomorphic functions are examined, which in turn point to new open problems. The book may be used by analysts for individual study or by graduate students.
Author: Filippo Bracci
Publisher: Springer Nature
ISBN: 3030367827
Category : Mathematics
Languages : en
Pages : 582
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The book faces the interplay among dynamical properties of semigroups, analytical properties of infinitesimal generators and geometrical properties of Koenigs functions. The book includes precise descriptions of the behavior of trajectories, backward orbits, petals and boundary behavior in general, aiming to give a rather complete picture of all interesting phenomena that occur. In order to fulfill this task, we choose to introduce a new point of view, which is mainly based on the intrinsic dynamical aspects of semigroups in relation with the hyperbolic distance and a deep use of Carathéodory prime ends topology and Gromov hyperbolicity theory. This work is intended both as a reference source for researchers interested in the subject, and as an introductory book for beginners with a (undergraduate) background in real and complex analysis. For this purpose, the book is self-contained and all non-standard (and, mostly, all standard) results are proved in details.
Author: Joseph Thomas Matti
Publisher:
ISBN:
Category : Analytic functions
Languages : en
Pages : 134
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Author: Hyungwoon Koo
Publisher:
ISBN:
Category : Holomorphic functions
Languages : en
Pages : 130
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Author: Ryan Keddie Tully-Doyle
Publisher:
ISBN: 9781339091945
Category :
Languages : en
Pages : 143
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Book Description
This dissertation concerns the investigation of function theoretic properties of certain classes of holomorphic functions in two or more variables by means of operator theoretic methods. Of primary concern will be the Schur class, the class of holomorphic functions from the complex polydisk into the complex unit disk, and the Pick class, the class of holomorphic functions from the complex poly-upperhalfplane into the complex upperhalfplane. In more than two variables, our results will concern certain large subclasses of these functions that satisfy an operator-theoretic condition analogous to a classical inequality of functions of one variable due to von Neumann [vN51]. These subclasses are typically referred to as the Schur-Agler subclass of the Schur functions (introduced in [Agl90]), and the Loewner subclass of the Pick functions (introduced in [AMY12b]. (In one or two variables, these subclasses coincide with the whole class.) These functions are amenable to investigation by means of an operator-theoretic construct called a Hilbert space model, introduced in [Agl90], which relates operator theoretic properties with function theoretic behavior. Hilbert space models are associated with and closely related to the notion of a transfer function realization from engineering and control theory [Hel87]. In Chapter 2, we describe a generalization of Hilbert space models for Schur functions on the bidisk that is well-suited to the investigation of boundary behavior of a function at a class of singular points for the function on the 2-torus. We prove that generalized models with certain regularity properties exist at these singularities. We then solve two function theoretic problems. First, we characterize the directional derivatives of a function in the Schur class at a singular point on the torus where a Caratheodory condition holds (following the generalization of the Julia-Carathedory theorem in [AMY12]. Second, we develop a representation theorem for functions in the two-variable Pick class analogous to the Nevanlinna representation theorem characterizing the Cauchy transforms of positive measures on the real line. In Chapter 3, we investigate more closely the structure of the generalized Hilbert space model. We characterize the directional derivatives in terms of a rational function depending on the structure of a positive contraction associated with a generalized model of a given Schur function. We describe classes of generalized models corresponding to different classes of singular points in the boundary for a Schur function in two variables. In Chapter 4, we generalize to several variables the Nevanlinna representation first investigated in Chapter 2. We show that for the Loewner class, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space that classify completely the Loewner class. We identify four types of such representations, and we obtain function-theoretic conditions that are necessary and sufficient for a given function to possess a representation of each of the four types.
Author: Kehe Zhu
Publisher: Springer Science & Business Media
ISBN: 0387275398
Category : Mathematics
Languages : en
Pages : 281
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Book Description
Can be used as a graduate text Contains many exercises Contains new results
Author: Mark Elin
Publisher: Springer Science & Business Media
ISBN: 3034605099
Category : Mathematics
Languages : en
Pages : 271
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Book Description
Linearization models for discrete and continuous time dynamical systems are the driving forces for modern geometric function theory and composition operator theory on function spaces. This book focuses on a systematic survey and detailed treatment of linearization models for one-parameter semigroups, Schröder’s and Abel’s functional equations, and various classes of univalent functions which serve as intertwining mappings for nonlinear and linear semigroups. These topics are applicable to the study of problems in complex analysis, stochastic and evolution processes and approximation theory.
