Pick Interpolation and Hilbert Function Spaces

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American Mathematical Soc., 2002 - Mathematics - 308 pages
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 $Hinfty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $Hinfty$ 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 theinterpolation 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.
 

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Contents

Prerequisites and Notation i
1
Introduction
7
Kernels and Function Spaces
15
Hardy Spaces
35
P2i
49
Pick Redux
55
Qualitative Properties of the Solution of the Pick Problem
71
Characterizing Kernels with the Complete Pick Property
79
Isometries
151
The Bidisk
167
The Extremal Three Point Problem on D2
195
Collections of Kernels
211
Function Spaces
237
Localization
263
Appendix A Schur Products
273
The Spectral Theorem for Normal mTuples
287

The Universal Pick Kernel
97
Interpolating Sequences
125

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