Orthogonal Polynomials on the Unit Circle: Spectral theory, Part 2
Presents an overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. This book discusses topics such as asymptotics of Toeplitz determinants (Szego's theorems), and limit theorems for the density of the zeros of orthogonal polynomials.
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Preface to Part 1
Rakhmanovs Theorem and Related Issues
Techniques of Spectral Analysis
The Basics 1
Periodic Verblunsky Coefficients
Spectral Analysis of Specific Classes
The Connection to Jacobi Matrices
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a.c. spectrum Aleksandrov analog analytic continuation argument bands bounded Cited CMV matrix compact subsets conjecture continuous function convergence Corollary define dense density of zeros Dirichlet data discuss eigenvalues eigenvector equation equivalent ergodic essential spectrum Example exists finite follows formula Furstenberg's theorem Geronimus given Hausdorff dimension Historical Notes holds implies integral invariant measure inverse isospectral Jacobi matrices Khrushchev Laurent polynomial Lebesgue Lemma limsup Lyapunov exponent Math meromorphic function minimal Caratheodory function Moreover nontrivial probability measure nonvanishing obeys OPRL OPUC orthogonal orthogonal polynomials periodic Verblunsky coefficients perturbations point masses Proof of Theorem Proposition prove pure point purely singular quadratic irrationality ratio asymptotics Remarks and Historical result Riemann surface Schrodinger operators Schur function Section selfadjoint sequence set of Verblunsky shows subshift Suppose theory torus uniformly on compact unitary Verblunsky coefficients Wall polynomial