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Product Code:  GSM/44.S 
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Softcover ISBN:  9781470468552 
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Softcover ISBN:  9781470468552 
Product Code:  GSM/44.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470420956 
Product Code:  GSM/44.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470468552 
eBook ISBN:  9781470420956 
Product Code:  GSM/44.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsGraduate Studies in MathematicsVolume: 44; 2002; 308 ppMSC: Primary 47; 30; 46; 32
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.
ReadershipGraduate students and research mathematicians interested in operator theory, function spaces, and analysis.

Table of Contents

Chapters

Chapter 0. Prerequisites and notation

Chapter 1. Introduction

Chapter 2. Kernels and function spaces

Chapter 3. Hardy spaces

Chapter 4. $P^2(\mu )$

Chapter 5. Pick redux

Chapter 6. Qualitative properties of the solution of the Pick problem in $H^\infty (\mathbb {D})$

Chapter 7. Characterizing kernels with the complete Pick property

Chapter 8. The universal Pick kernel

Chapter 9. Interpolating sequences

Chapter 10. Model theory I: Isometries

Chapter 11. The bidisk

Chapter 12. The extremal three point problem on $\mathbb {D}^2$

Chapter 13. Collections of kernels

Chapter 14. Model theory II: Function spaces

Chapter 15. Localization

Appendix A. Schur products

Appendix B. Parrott’s lemma

Appendix C. Riesz interpolation

Appendix D. The spectral theorem for normal $m$tuples


Reviews

Written in a clear, straightforward style, at a level to make it accessible to someone—a midlevel graduate student, say—who wishes to study the material in detail for the first time ... contains exercises ... as well as ... open questions. It brings the reader up to the current ‘state of the art’ and so will be a valuable resource for the specialist ... would be an excellent basis for a graduate seminar or topics course.
Mathematical Reviews 
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.
Palle Jorgensen


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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.
Graduate students and research mathematicians interested in operator theory, function spaces, and analysis.

Chapters

Chapter 0. Prerequisites and notation

Chapter 1. Introduction

Chapter 2. Kernels and function spaces

Chapter 3. Hardy spaces

Chapter 4. $P^2(\mu )$

Chapter 5. Pick redux

Chapter 6. Qualitative properties of the solution of the Pick problem in $H^\infty (\mathbb {D})$

Chapter 7. Characterizing kernels with the complete Pick property

Chapter 8. The universal Pick kernel

Chapter 9. Interpolating sequences

Chapter 10. Model theory I: Isometries

Chapter 11. The bidisk

Chapter 12. The extremal three point problem on $\mathbb {D}^2$

Chapter 13. Collections of kernels

Chapter 14. Model theory II: Function spaces

Chapter 15. Localization

Appendix A. Schur products

Appendix B. Parrott’s lemma

Appendix C. Riesz interpolation

Appendix D. The spectral theorem for normal $m$tuples

Written in a clear, straightforward style, at a level to make it accessible to someone—a midlevel graduate student, say—who wishes to study the material in detail for the first time ... contains exercises ... as well as ... open questions. It brings the reader up to the current ‘state of the art’ and so will be a valuable resource for the specialist ... would be an excellent basis for a graduate seminar or topics course.
Mathematical Reviews 
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.
Palle Jorgensen