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Softcover ISBN:  9780821849330 
Product Code:  SURV/92.S 
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Book DetailsMathematical Surveys and MonographsVolume: 92; 2002; 461 ppMSC: Primary 47; 30; 93
Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS, 2002, this unique work combines four major topics of modern analysis and its applications:
A. Hardy classes of holomorphic functions,
B. Spectral theory of Hankel and Toeplitz operators,
C. Function models for linear operators and free interpolations, and
D. Infinitedimensional system theory and signal processing.
This volume contains Parts A and B.
Hardy classes of holomorphic functions is known to be the most powerful tool in complex analysis for a variety of applications, starting with Fourier series, through the Riemann \(\zeta\)function, all the way to Wiener's theory of signal processing.
Spectral theory of Hankel and Toeplitz operators becomes the supporting pillar for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, NevanlinnaPick and Carathéodory interpolation, and the best rational approximations are considered to illustrate the power of Hankel and Toeplitz operators.
The book is geared toward a wide audience of readers, from graduate students to professional mathematicians, interested in operator theory and functions of a complex variable. The two volumes develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications.
ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Volume 1. Hardy, Hankel and Toeplitz

Part A. An invitation to Hardy Classes

1. Invariant subspaces of $L^2(\mu )$

2. First applications

3. $H^p$ classes. Canonical factorization

4. Szegö infimum, and generalized Phragmén–Lindelöf principle

5. Harmonic analysis in $L^2(\mathbb {T},\mu )$

6. Transfer to the halfplane

7. Timeinvariant filtering

8. Distance formulae and zeros of the Riemann $\zeta $function

Part B. Hankel and Toeplitz operators

1. Hankel operators and their symbols

2. Compact Hankel operators

3. Applications to Nevanlinna–Pick interpolation

4. Essential spectrum. The first step: Elements of Toeplitz operators

5. Essential spectrum. The second step: The Hilbert matrix and other Hankel operators

6. Hankel and Toeplitz operators associated with moment problems

7. Singular numbers of Hankel operators

8. Trace class Hankel operators

9. Inverse spectral problems, stochastic processes and onesided invertibility


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Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS, 2002, this unique work combines four major topics of modern analysis and its applications:
A. Hardy classes of holomorphic functions,
B. Spectral theory of Hankel and Toeplitz operators,
C. Function models for linear operators and free interpolations, and
D. Infinitedimensional system theory and signal processing.
This volume contains Parts A and B.
Hardy classes of holomorphic functions is known to be the most powerful tool in complex analysis for a variety of applications, starting with Fourier series, through the Riemann \(\zeta\)function, all the way to Wiener's theory of signal processing.
Spectral theory of Hankel and Toeplitz operators becomes the supporting pillar for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, NevanlinnaPick and Carathéodory interpolation, and the best rational approximations are considered to illustrate the power of Hankel and Toeplitz operators.
The book is geared toward a wide audience of readers, from graduate students to professional mathematicians, interested in operator theory and functions of a complex variable. The two volumes develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications.
Graduate students and research mathematicians interested in analysis.

Volume 1. Hardy, Hankel and Toeplitz

Part A. An invitation to Hardy Classes

1. Invariant subspaces of $L^2(\mu )$

2. First applications

3. $H^p$ classes. Canonical factorization

4. Szegö infimum, and generalized Phragmén–Lindelöf principle

5. Harmonic analysis in $L^2(\mathbb {T},\mu )$

6. Transfer to the halfplane

7. Timeinvariant filtering

8. Distance formulae and zeros of the Riemann $\zeta $function

Part B. Hankel and Toeplitz operators

1. Hankel operators and their symbols

2. Compact Hankel operators

3. Applications to Nevanlinna–Pick interpolation

4. Essential spectrum. The first step: Elements of Toeplitz operators

5. Essential spectrum. The second step: The Hilbert matrix and other Hankel operators

6. Hankel and Toeplitz operators associated with moment problems

7. Singular numbers of Hankel operators

8. Trace class Hankel operators

9. Inverse spectral problems, stochastic processes and onesided invertibility