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 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.
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Acknowledgements
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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Mashreghi, J., Ransford, T. (2018). Approximation in the Closed Unit Ball. In: Mashreghi, J., Manolaki, M., Gauthier, P. (eds) New Trends in Approximation Theory. Fields Institute Communications, vol 81. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7543-3_5
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