Orthogonal polynomials on the unit circle

Summary

Orthogonal polynomials on the unit circle are the polynomials obtained by Gram–Schmidt orthogonalization of $1, z, z^2, \dots$ in $L^2$ of a probability measure on the unit circle. They are the circle analogue of orthogonal polynomials on the real line. Instead of a three-term recurrence they satisfy the Szegő recurrence, governed by a sequence of numbers in the unit disk, the Verblunsky coefficients. The theory, started by Gábor Szegő, is linked to Toeplitz matrices, to prediction theory of stationary processes, and to spectral theory of CMV matrices.