Read e-book online Complex Convexity and Analytic Functionals PDF

By Mats Andersson

ISBN-10: 3034878710

ISBN-13: 9783034878715

ISBN-10: 3034896050

ISBN-13: 9783034896054

A set in complicated Euclidean area is termed C-convex if all its intersections with advanced traces are contractible, and it truly is acknowledged to be linearly convex if its supplement is a union of complicated hyperplanes. those notions are intermediates among usual geometric convexity and pseudoconvexity. Their significance was once first manifested within the pioneering paintings of André Martineau from approximately 40 years in the past. considering the fact that then quite a few new comparable effects were received through many various mathematicians. the current publication places the fashionable idea of complicated linear convexity on a fantastic footing, and provides an intensive and up to date survey of its present prestige. purposes comprise the Fantappié transformation of analytic functionals, crucial illustration formulation, polynomial interpolation, and options to linear partial differential equations.

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To see this we use the non-degeneracy of E to find a point Aa E Ca \ E. p~(0) :::; 41AI. p~(0) :::; C < 00. 8 we can not assume the sets E a to be uniformly bounded. 7]. pa. : 1, we conclude that 'ljJ is injective. pa. Let U be a connected open relatively compact subset of E a containing the origin. Then U a is relatively compact in E, and it follows that any nearby set U aj is also contained in E, which amounts to U C E aj . paj are defined in U with values in D, so some subsequence of them converges to a limit function -J; with the property -J;(O) = 0, and hence -J;(U) c D.

0 The proof of the preceding lemma becomes much simpler if we impose the extra condition that E be polynomially convex. Let U be any polynomially convex neighborhood U of E. Then each intersection un£ with a complex line £ has simply connected components. With the notation from the proof, the set Uf is therefore just the component of £ n U that contains a, and the union U of all Uf is a starlike open neighborhood of E which is contained in U. 3 C-convexity: Duality and invariance We are now about to establish some more delicate properties of C-convex sets, much in analogy to those of ordinary convex sets presented in Chapter 1.

Proof. There is no loss of generality in assuming 0 E E C cn and we consider first the non-degenerate case, that is, no affine line is contained in E. In order to define 'P we associate to each intersection En Ca the planar domain E a = {t E O. 4. p: B n -+ E. p is in fact a homeomorphism. To this end we first observe that the linear convexity of E implies the following semi-continuity property of the set-valued mapping a ~ E a : For any point A rt E a and any sequence aj -+ a one can find a sequence Aj -+ A with Aj rt E aj .

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Complex Convexity and Analytic Functionals by Mats Andersson

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