Approximation in the Closed Unit Ball

  • Javad Mashreghi
  • Thomas RansfordEmail author
Part of the Fields Institute Communications book series (FIC, volume 81)


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 Carathéodory, 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.


Approximation Unit ball Blaschke product Inner function Convex hull 

2010 Mathematics Subject Classification.

30J05 30J10 



Javad Mashreghi was supported by a grant from NSERC. Thomas Ransford was supported by grants from NSERC and the Canada research chairs program.


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Authors and Affiliations

  1. 1.Département de Mathématiques et de StatistiqueUniversité LavalQuébecCanada

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