1. Approximation in the Closed Unit Ball
- Author
-
Javad Mashreghi and Thomas Ransford
- Subjects
Convex hull ,Discrete mathematics ,Unit sphere ,Mathematics::Complex Variables ,Blaschke product ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,Closure (topology) ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Simple (abstract algebra) ,symbols ,0101 mathematics ,Numerical range ,Mathematics - Abstract
In this expository article, we present a number of classic theorems that serve to identify the closure in the sup-norm of various sets of Blaschke products, inner functions and their quotients, as well as the closure of the convex hulls of these sets. The results presented include theorems of Caratheodory, Fisher, Helson–Sarason, Frostman, Adamjan–Arov–Krein, Douglas–Rudin and Marshall. As an application of some of these ideas, we obtain a simple proof of the Berger–Stampfli spectral mapping theorem for the numerical range of an operator.
- Published
- 2018