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Full Description
This unique work combines together in two volumes four formally distinct topics of modern analysis and its applications: Hardy classes of holomorphic functions; spectral theory of Hankel and Toeplitz operators; function models for linear operators and free interpolations; and, infinite-dimensional system theory and signal processing. This volume, Volume 1, contains Parts A and B; Volume 2, Model Operators and Systems, contains Parts C and D. Hardy classes of holomorphic functions: this topic is known to be the most powerful tool of 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: these now become the supporting pillars 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.
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. These volumes are geared toward a wide audience of readers, from graduate students to professional mathematicians. They develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications.
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 Harmonc 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.
