Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory
This two-volume monograph is a comprehensive treatment of the theory of probability measures on the unit circle, viewed from the perspective of orthogonal polynomials defined by these measures. Part I primarily discusses main topics in the subject between 1920 and 1985, with the addition of the CMV (Cantero, Moral and Velásquez) matrix representation. Part II deals with the presentation of the theory of orthogonal polynomials on the unit circle as a spectral theory problem analogous to spectral theory for Schrödinger operators or Jacobi matrices.
The book establishes a connection between Verblunsky coefficients (coefficients of the recurrence equation for orthogonal polynomials) and measures, an analogue of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed here, the reader can find a study of asymptotics of Toeplitz determinants, limit theorems for the density of zeros of orthogonal polynomials, matrix representations for multiplications by CMV matrices, periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between theories of orthogonal polynomials on the unit circle and on the real line. Summarizing, the book is intended as a companion to basic literature on orthogonal polynomials on the unit circle. It consists of 13 chapters with remarks, historical notes and some appendices. The reference list contains more than one thousand references. The reviewer is convinced that the book will offer an inspiration for further research. It can be strongly recommended to mathematicians specializing in the theory of Schrödinger operators and the theory of orthogonal polynomials.