Pick Interpolation and Hilbert Function Spaces by Jim Agler (PDF)

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.

User’s Reviews

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⭐Classical and modern.The “interpolation” part of the title refers to a classical result of G. Pick [1916], and R. Nevanlinna [1919]. They did it independently. Since the disk D (in one complex variable)is a model of a hyperbolic metric space, interpolation can’t be done by just drawing straight lines. If two finite sets of points in D are given(same number of points in the two sets), the issue is to find an analytic function mapping D into itself, which at the same time matches up the points in one set with those in the other. The necessary and sufficient condition for the existence, and the solution, when it exists, is the classical issue. Since this, the framework has offered a wonderful testing ground for main trends in modern analysis, such as the theory of operators in Hilbert space, and for modern techniques in systems theory of engineering, and of scattering theory. The material is wonderfully presented, and the book serves as a lovely introduction to the subject. It is written by two authorities in the field, and helps grad students get entry into an exciting, modern, and very active research area.

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