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Author: Steven Gubkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 78
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Book Description
An "approximation theorem" usually establishes the density of one space of functions in another, with respect to some norm. Classical examples include Runge's theorem and Mergelyan's theorem from single variable complex analysis. One analogue of Runge's theorem to several complex variables is the Oka-Weil Theorem. We offer an apparently new proof of this theorem which avoids a tricky duality argument. The main new theorems in the thesis are analogues of Mergelyan's theorem, only using L2 instead of uniform norm approximations. In particular we show that if an open set U is strictly hyper convex, the space of holomorphic functions defined in a neighborhood of U is dense with respect to L2 norm in the space of square integrable holomorphic functions defined on U. We then extend this result to L2 approximation of dbar-closed (p,q)-forms.
Author: Steven Gubkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 78
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Book Description
An "approximation theorem" usually establishes the density of one space of functions in another, with respect to some norm. Classical examples include Runge's theorem and Mergelyan's theorem from single variable complex analysis. One analogue of Runge's theorem to several complex variables is the Oka-Weil Theorem. We offer an apparently new proof of this theorem which avoids a tricky duality argument. The main new theorems in the thesis are analogues of Mergelyan's theorem, only using L2 instead of uniform norm approximations. In particular we show that if an open set U is strictly hyper convex, the space of holomorphic functions defined in a neighborhood of U is dense with respect to L2 norm in the space of square integrable holomorphic functions defined on U. We then extend this result to L2 approximation of dbar-closed (p,q)-forms.
Author: Takeo Ohsawa
Publisher: Springer
ISBN: 4431568522
Category : Mathematics
Languages : en
Pages : 258
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Book Description
This monograph presents the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Special emphasis is put on the new precise results on the L2 extension of holomorphic functions in the past 5 years.In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and separately by Q.-A. Guan and X.-Y. Zhou. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, Guan–Zhou, and Berndtsson–Lempert. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L2 method obtained during the past 15 years.
Author: Daniel Breaz
Publisher: Springer Nature
ISBN: 3030401200
Category : Mathematics
Languages : en
Pages : 538
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Book Description
The contributions to this volume are devoted to a discussion of state-of-the-art research and treatment of problems of a wide spectrum of areas in complex analysis ranging from pure to applied and interdisciplinary mathematical research. Topics covered include: holomorphic approximation, hypercomplex analysis, special functions of complex variables, automorphic groups, zeros of the Riemann zeta function, Gaussian multiplicative chaos, non-constant frequency decompositions, minimal kernels, one-component inner functions, power moment problems, complex dynamics, biholomorphic cryptosystems, fermionic and bosonic operators. The book will appeal to graduate students and research mathematicians as well as to physicists, engineers, and scientists, whose work is related to the topics covered.
Author: Marek Jarnicki
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110627698
Category : Mathematics
Languages : en
Pages : 455
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Book Description
This second extended edition of the classic reference on the extension problem of holomorphic functions in pluricomplex analysis contains a wealth of additional material, organized under the original chapter structure, and covers in a self-contained way all new and recent developments and theorems that appeared since the publication of the first edition about twenty years ago.
Author: Herbert Alexander
Publisher: Springer Science & Business Media
ISBN: 0387225862
Category : Mathematics
Languages : en
Pages : 265
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Book Description
A development of some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader, and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. This third edition contains new material on maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter commenting on history and recent developments, as well as an updated and expanded reading list.
Author: Jiri Lebl
Publisher: Lulu.com
ISBN: 1365095576
Category :
Languages : en
Pages : 142
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Book Description
This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014 and Spring 2016 semester at Oklahoma State University. The course covers basics of holomorphic function theory, CR geometry, the dbar problem, integral kernels and basic theory of complex analytic subvarieties. See http: //www.jirka.org/scv/ for more information.
Author: Simon Foucart
Publisher: Springer Nature
ISBN: 3031664973
Category :
Languages : en
Pages : 294
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Book Description
Author: Frederick P. Greenleaf
Publisher: American Mathematical Soc.
ISBN: 0821850342
Category : Mathematics
Languages : en
Pages : 312
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Book Description
Contains papers presented at the conference on Banach Algebras and Several Complex Variables held June 21-24, 1983, to honor Professor Charles E Rickart upon his retirement from Yale University. This work includes articles that present advances in topics related to Banach algebras, function algebras and infinite dimensional holomorphy.
Author: L. Hormander
Publisher: Elsevier
ISBN: 0444105239
Category : Mathematics
Languages : en
Pages : 227
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Book Description
An Introduction to Complex Analysis in Several Variables
Author: Carlos A. Berenstein
Publisher: Springer Science & Business Media
ISBN: 9780387973494
Category : Mathematics
Languages : en
Pages : 694
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Book Description
This text gives an overview of the basic properties of holomorphic functions of one complex variable. Topics studied in this overview include a detailed description of differential forms, homotopy theory, and homology theory, as the analytic properties of holomorphic functions, the solvability of the inhomogeneous Cauchy-Riemann equation with emphasis on the notation of compact families, the theory of growth of subharmonic functions, and an introduction to the theory of sheaves, covering spaces and Riemann surfaces. To further illuminate the material, a large number of exercises of differing levels of difficulty have been added.
