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Operators, Functions, and Systems Hardy, Hankel, and Toeplitz An Easy Reading by Nikolai K. Nikolski

Operators, Functions, and Systems Hardy, Hankel, and Toeplitz An Easy Reading

Part of the Mathematical Surveys and Monographs Series


Operators, Functions, and Systems Hardy, Hankel, and Toeplitz An Easy Reading by Nikolai K. Nikolski

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. Infinite-dimensional 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, Nevanlinna-Pick and Caratheodory 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. Table of Contents: An invitation to Hardy classes/Contents: Foreword to Part A; Invariant subspaces of $L^2(\mu)$; First applications; $H^p$ classes. Canonical factorization; Szego infimum, and generalized Phragmen-Lindelof principle; Harmonic analysis in $L^2(\mathbb{T},\mu)$; Transfer to the half-plane; Time-invariant filtering; Distance formulae and zeros of the Riemann $\zeta$-function. Hankel and Toeplitz operators/Contents: Foreword to Part B; Hankel operators and their symbols; Compact Hankel operators; Applications to Nevanlinna-Pick interpolation; Essential spectrum. The first step: Elements of Toeplitz operators; Essential spectrum. The second step: The Hilbert matrix and other Hankel operators; Hankel and Toeplitz operators associated with moment problems; Singular numbers of Hankel operators; Trace class Hankel operators; Inverse spectral problems, stochastic processes and one-sided invertibility; Bibliography; Author index; Subject index; Symbol index. This is a reprint of the 2002 original. (SURV/92.S)

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Book Info

Publication date

15th January 2002


Nikolai K. Nikolski

More books by Nikolai K. Nikolski
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American Mathematical Society


461 pages


Mathematical theory of computation
Differential & Riemannian geometry
Systems analysis & design
Complex analysis, complex variables



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