Polynomial Convexity

ISBN-13: 9780817645373 / Angielski / Twarda / 2007 / 439 str.

Edgar Lee Stout

Opis

This book is devoted to an exposition of the theory of polynomially convex sets.Acompact N subset of C is polynomially convex if it is de?ned by a family, ?nite or in?nite, of polynomial inequalities. These sets play an important role in the theory of functions of several complex variables, especially in questions concerning approximation. On the one hand, the present volume is a study of polynomial convexity per se, on the other, it studies the application of polynomial convexity to other parts of complex analysis, especially to approximation theory and the theory of varieties. N Not every compact subset of C is polynomially convex, but associated with an arbitrary compact set, say X, is its polynomially convex hull, X, which is the intersection of all polynomially convex sets that contain X. Of paramount importance in the study of polynomial convexity is the study of the complementary set X X. The only obvious reason for this set to be nonempty is for it to have some kind of analytic structure, and initially one wonders whether this set always has complex structure in some sense. It is not long before one is disabused of this naive hope; a natural problem then is that of giving conditions under which the complementary set does have complex structure. In a natural class of one-dimensional examples, such analytic structure is found. The study of this class of examples is one of the major directions of the work at hand.

Recenzje

From the reviews:

"The style is rigorous, elegant and clear, the exposition is beautiful. The book is an extremely important tool to every researcher interested in the subject, as it contains basic facts and therefore will remain a standard reference in the future and, moreover, it opens a perspective on further directions of research."-Zentralblatt Math

"This is an excellent ... introductory book for researchers in complex function theory and approximation theory that certainly becomes one of the chief references for these topics. I think the importance and main techniques of how to use polynomial convexity as a standard tool is rather clear for the mentioned experts. ... The book is highly recommended for every researcher and postgraduate student working in areas with intensive use of complex analysis, analytic varieties or approximation theory." (László Stachó, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

"Polynomial convexity is an important concept in the theory of functions of several complex variables, especially for approximation. This excellent exposition of a rich theory presents the general properties of polynomially convex sets with attention to hulls of one-dimensional sets ... . Together with the comprehensive bibliography and the numerous interesting historical remarks this book will serve as a standard reference for many years." (F. Haslinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

Spis treści

Preface. Introduction. Polynomial convexity. Uniform algebras. Plurisubharmonic fuctions. The Cauchy-Fantappiè Integral. The Oka—Weil Theorem. Some examples. Hulls with no analytic structure.- Some General Properties of Polynomially Convex Sets. Applications of the Cousin problems. Two characterizations of polynomially convex sets. Applications of Morse theory and algebraic topology. Convexity in Stein manifolds.- Sets of Finite Length. Introduction. One-dimensional varieties. Geometric preliminaries. Function-theoretic preliminaries. Subharmonicity results. Analytic structure in hulls. Finite area. The continuation of varieties.- Sets of Class A1. Introductory remarks. Measure-theoretic preliminaries. Sets of class A1. Finite area. Stokes’s Theorem. The multiplicity function. Counting the branches.- Further Results. Isoperimetry. Removable singularities. Surfaces in strictly pseudoconvex boundaries.- Approximation. Totally real manifolds. Holomorphically convex sets. Approximation on totally real manifolds. Some tools from rational approximation. Algebras on surfaces. Tangential approximation.- Varieties in Strictly Pseudoconvex Domains. Interpolation. Boundary regularity. Uniqueness.- Examples and Counter Examples. Unions of planes and balls. Pluripolar graphs. Deformations. Sets with symmetry.- Bibliography. Index.

Blurb

This comprehensive monograph is devoted to the study of polynomially convex sets, which play an important role in the theory of functions of several complex variables.

Important features of Polynomial Convexity:

*Presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets.

*Motivates the theory with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

*Examines in considerable detail questions of uniform approximation, especially on totally real sets, for the most part on compact sets but with some attention to questions of global approximation on noncompact sets.

*Discusses important applications, e.g., to the study of analytic varieties and to the theory of removable singularities for CR functions.

*Requires of the reader a solid background in real and complex analysis together with some previous experience with the theory of functions of several complex variables as well as the elements of functional analysis.

This beautiful exposition of a rich and complex theory, which contains much material not available in other texts, is destined to be the standard reference for many years, and will appeal to all those with an interest in multivariate complex analysis.