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 .
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-Nikodym and 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 monic orthogonal Szegà  polynomials satisfy a recurrence relation of the form
for 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 states that for any sequence of numbers in there is a unique nontrivial probability measure on with .
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.
For any nontrivial probability measure on , Verblunsky's form of Szegà Â's theorem states that
The left-hand side is independent of but unlike Szegà Â'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.
Rakhmanov's theorem states that if the absolutely continuous part of the measure is positive almost everywhere then the Verblunsky coefficients tend to 0.
The RogersâÂÂSzegà  polynomials are an example of orthogonal polynomials on the unit circle.