Problems in Function Theory

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Abstract: Garnett will work on several problems in classical one dimensional complex analysis. The first problem is to approximate any Blaschke product uniformly on the open disc by Blaschke products whose zeros are sufficiently spread apart and thin that the corresponding Riesz mass is bounded in all holomorphic coordinate systems (i. e. is a Carleson measure) The approximation should be effected using explicit constructions. The second problem is to exhibit large compact sets whose complements do not support nonconstant bounded analytic functions. The third problem is a corona problem for infinitely connected plane domains whose boundaries lie on rectifiable Jordan curves. It too requires some new explicit constructions. The fourth problem is to show that a Jordan curve depends continuously on its welding map, which is the correspondence on the unit circle that connects the conformal mapping of the inside of the disc to one side of the curve and the outside of the disc to the other side of the curve. The methods to be used on these problems will be constructive so that they can be give explicit computer aided constructions of analytic functions and conformal mappings. Analytic functions and conformal mappings have broad applications in fluid dynamics, acoustics, and electrical engineering, and in these applications constructions are more useful than general existence theorems.