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

Suppose that  is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to  are the polynomials  with leading term  that are orthogonal with respect to the measure . The Szegő recurrence 

The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form 

for  and initial condition , with

and constants  given by

called the Verblunsky coefficients. Subsequently, Geronimus' theorem states that the Verblunsky coefficients associated with  are the Schur parameters:

Verblunsky's theorem

Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support. 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

Verblunsky's form of Szegő's theorem states that 
 
where  is the absolutely continuous part of the measure . Verblunsky's form also allows for a non-zero singular part while  in Szegő's original version. Rakhmanov's theorem

Rakhmanov's theorem states that if the absolutely continuous part  of the measure  is positive almost everywhere then the Verblunsky coefficients  tend to 0. Examples

The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle. See also
 Schur class

References

Orthogonal polynomials