Author	Andersson, Mats. author
Title	Complex Convexity and Analytic Functionals [electronic resource] / by Mats Andersson, Ragnar Sigurdsson, Mikael Passare
Imprint	Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2004
Connect to	http://dx.doi.org/10.1007/978-3-0348-7871-5
Descript	XI, 164 p. online resource

A set in complex Euclidean space is called C-convex if all its intersections with complex lines are contractible, and it is said to be linearly convex if its complement is a union of complex hyperplanes. These notions are intermediates between ordinary geometric convexity and pseudoconvexity. Their importance was first manifested in the pioneering work of Andrรฉ Martineau from about forty years ago. Since then a large number of new related results have been obtained by many different mathematicians. The present book puts the modern theory of complex linear convexity on a solid footing, and gives a thorough and up-to-date survey of its current status. Applications include the Fantappiรฉ transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations

1 Convexity in Real Projective Space -- 1.1 Convexity in real affine space -- 1.2 Real projective space -- 1.3 Convexity in real projective space -- 2 Complex Convexity -- 2.1 Linearly convex sets -- 2.2 ?-convexity: Definition and examples -- 2.3 ?-convexity: Duality and invariance -- 2.4 Open ?-convex sets -- 2.5 Boundary properties of ?-convex sets -- 2.6 Spirally connected sets -- 3 Analytic Functionals and the Fantappiรจ Transformation -- 3.1 The basic pairing in affine space -- 3.2 The basic pairing in projective space -- 3.3 Analytic functionals in affine space -- 3.4 Analytic functionals in projective space -- 3.5 The Fantappiรจ transformation -- 3.6 Decomposition into partial fractions -- 3.7 Complex Kergin interpolation -- 4 Analytic Solutions to Partial Differential Equations -- 4.1 Solvability in ?-convex sets -- 4.2 Solvability and P-convexity for carriers -- References