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Full Description
The book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $H^\infty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $H^\infty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces.
The authors then consider the interpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem, and the hereditary functional calculus.
Contents
Prerequisites and notation
Introduction
Kernels and function spaces
Hardy spaces
$P^2(\mu)$
Pick redux
Qualitative properties of the solution of the Pick problem in $H^\infty(\mathbb{D})$
Characterizing kernels with the complete Pick property
The universal Pick kernel
Interpolating sequences
Model theory I: Isometries
The bidisk
The extremal three point problem on $\mathbb{D}^2$
Collections of kernels
Model theory II: Function spaces
Localization
Schur products
Parrott's lemma
Riesz interpolation
The spectral theorem for normal $m$-tuples
Bibliography
Index
