Orthogonal polynomials on the unit circle
In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by.
Definition
Let be a probability measure on the unit circle and assume is nontrivial, i.e., its support is an infinite set. By a combination of the Radon-Nikodymand Lebesgue decomposition theorems, any such measure can be uniquely
decomposed into
where is singular with respect to and with the absolutely continuous part of.
The orthogonal polynomials associated with are defined as
such that
The Szegő recurrence
The monic orthogonal Szegő polynomials satisfy a recurrence relation of the formfor and initial condition, with
and constants in the open unit disk given by
called the Verblunsky coefficients. Moreover,
Geronimus' theorem states that the Verblunsky coefficients associated with are the Schur parameters:
Verblunsky's theorem
Verblunsky's theorem states that for any sequence of numbers in there is a unique nontrivial probability measure on with.Baxter's theorem
Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of form an absolutely convergent series and the weight function is strictly positive everywhere.Szegő's theorem
For any nontrivial probability measure on, Verblunsky's form of Szegő's theorem states thatThe left-hand side is independent of but unlike Szegő's original version, where, Verblunsky's form does allow. Subsequently,
One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.