**Mathematical Surveys and Monographs**

Volume: 93;
2002;
439 pp;
Softcover

MSC: Primary 47; 30; 93;

Print ISBN: 978-0-8218-5265-1

Product Code: SURV/93.S

List Price: $108.00

Individual Member Price: $86.40

**Electronic ISBN: 978-1-4704-1320-0
Product Code: SURV/93.S.E**

List Price: $108.00

Individual Member Price: $86.40

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# Operators, Functions, and Systems: An Easy Reading: Volume 2: Model Operators and Systems

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*Nikolai K. Nikolski*

Together with the companion volume by the same author, Operators,
Functions, and Systems: An Easy Reading. Volume 1: Hardy, Hankel, and
Toeplitz, Mathematical Surveys and Monographs, Vol. 92, 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. Infinite-dimensional system theory and signal
processing.

This volume contains Parts C and D.

Function models for linear operators and free interpolations:
This is a universal topic and, indeed, is the most influential operator
theory technique in the post-spectral-theorem era. In this book, its
capacity is tested by solving generalized Carleson-type interpolation
problems.

Infinite-dimensional system theory and signal processing: This
topic is the touchstone of the three previously developed techniques. The
presence of this applied topic in a pure mathematics environment reflects
important changes in the mathematical landscape of the last 20 years, in
that the role of the main consumer and customer of harmonic, complex, and
operator analysis has more and more passed from differential equations,
scattering theory, and probability to control theory and signal
processing.

This and the companion volume are geared toward a wide audience of
readers, from graduate students to professional mathematicians. They
develop an elementary approach to the subject while retaining an expert
level that can be applied in advanced analysis and selected applications.

#### Readership

Graduate students and research mathematicians interested in operator theory, functions of a complex variable, systems theory, and control.

#### Table of Contents

# Table of Contents

## Operators, Functions, and Systems: An Easy Reading: Volume 2: Model Operators and Systems

- Contents ix10
- A Few Words about the Book v6 free
- Part C. Model Operators and Free Interpolation 116 free
- Part D. Analytic Problems in Linear System Theory 197212
- Foreword to Part D 201216
- Chapter 1. Basic Theory 205220
- Chapter 2. First Optimizations: Multiplicity of the Spectrum and the DISC 237252
- Chapter 3. Eigenvector Decompositions, Vector Valued Exponentials, and Squared Optimization 267282
- 3.1. Examples of Parabolic and Hyperbolic Systems 268283
- 3.2. Complete Generators 271286
- 3.3. Riesz Bases and Exact Controllability 274289
- 3.4. Generalized Controllability and Renormalizations 277292
- 3.5. Null Controllability (NCO) 284299
- 3.6. Weak Controllability 289304
- 3.7. Squared (Energy) Optimization 290305
- 3.8. Control at Time τ = ∞ and Interpolation in H[sup(2)] 217
- 3.9. Notes and Remarks 295310

- Chapter 4. A Glance at Bases of Exponentials and of Reproducing Kernels 299314
- 4.1. Small Perturbations of Harmonic Frequencies 300315
- 4.2. Bases of Exponentials on the Half-Line 301316
- 4.3. Bases of Exponentials on Finite Intervals 304319
- 4.4. Bases of Reproducing Kernels in Model Spaces 307322
- 4.5. Back to Exponentials 313328
- 4.6. A Levinson Completeness Theorem 316331
- 4.7. Exercises and Further Results 321336
- 4.8. Notes and Remarks 334349

- Chapter 5. A Brief Introduction to H[sup(∞)] Control 343358
- 5.1. Input-Output Maps and Transfer Functions 343358
- 5.2. Noise Minimization, Feedback Control, and Sensitivity 346361
- 5.3. Remarks on Robust Stabilization 350365
- 5.4. Scattering Type Input-Output, and Hankel Operators 350365
- 5.5. Remarks on Finite Dimensional Systems 355370
- 5.6. Exercises and Further Results 356371
- 5.7. Notes and Remarks 358373

- Bibliography 361376
- Author Index 401416
- Subject Index 411426
- Symbol Index 435450
- Errata to Volume 1 439454