Description
BIRKHAUSER BOSTON INC Polynomial Convexity by Edgar Lee Stout
This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries._x000D_ _x000D_
Preface. _x000D_
Introduction. Polynomial convexity. Uniform algebras. Plurisubharmonic fuctions. The Cauchy-Fantappie Integral. The Oka-Weil Theorem. Some examples. Hulls with no analytic structure.- _x000D_
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.- _x000D_
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.- _x000D_
Sets of Class A1. Introductory remarks. Measure-theoretic preliminaries. Sets of class A1. Finite area. Stokes's Theorem. The multiplicity function. Counting the branches.- _x000D_
Further Results. Isoperimetry. Removable singularities. Surfaces in strictly pseudoconvex boundaries.- _x000D_
Approximation. Totally real manifolds. Holomorphically convex sets. Approximation on totally real manifolds. Some tools from rational approximation. Algebras on surfaces. Tangential approximation.- _x000D_
Varieties in Strictly Pseudoconvex Domains. Interpolation. Boundary regularity. Uniqueness.-_x000D_
Examples and Counter Examples. Unions of planes and balls. Pluripolar graphs. Deformations. Sets with symmetry.- _x000D_
Bibliography. Index._x000D_