Mats Andersson Livres

This book stems from lectures at Chalmers University of Technology and Goteborg University over the past decade. Unlike most introductory texts on complex analysis, it assumes the reader has a foundation in basic real analysis, allowing for a swift exploration of fundamental concepts. This approach facilitates coverage of classical highlights, including Fatou theorems and aspects of Nevanlinna theory, as well as more contemporary topics like the corona theorem and HI_BMO duality, all within a one-semester course framework. Certain sections, specifically in Chapters 2, 3, 5, and 7, introduce special topics not included in the original lecture notes; these can be omitted without disrupting the flow of the material. The organization of topics reflects historical developments, with the first five chapters focusing on 19th-century theories, while later chapters address advancements from the early 20th century through the 1980s. The selection of presentation methods and proofs is approached with care, aiming to highlight connections to real and harmonic analysis while maintaining a focus on classical complex function theory.

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 André 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 Fantappié transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations.