This two-part volume gives a comprehensive
overview of the theory of probability measures on the unit circle,
viewed especially in terms of the orthogonal polynomials defined by
those measures. A major theme involves the connections between the
Verblunsky coefficients (the coefficients of the recurrence equation
for the orthogonal polynomials) and the measures, an analog of the
spectral theory of one-dimensional Schrödinger operators.

Among
the topics discussed along the way are the asymptotics of Toeplitz
determinants (Szegő's theorems), limit theorems for the density of the
zeros of orthogonal polynomials, matrix representations for
multiplication by
(CMV matrices), periodic Verblunsky coefficients from the point of
view of meromorphic functions on hyperelliptic surfaces, and
connections between the theories of orthogonal polynomials on the unit
circle and on the real line.

The book is suitable for graduate students and researchers interested in analysis.

Barry Simon