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 Szegő (1920, 1921, 1939).
be a probability measure on the unit circle
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
d θ
the absolutely continuous part of
[1] The orthogonal polynomials associated with
are defined as such that The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form for
and initial condition
in the open unit disk
given by called the Verblunsky coefficients.
[2] Moreover, Geronimus' theorem states that the Verblunsky coefficients associated with
are the Schur parameters:[3] Verblunsky's theorem states that for any sequence of numbers
there is a unique nontrivial probability measure
[4] 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.
[5] For any nontrivial probability measure
, Verblunsky's form of Szegő's theorem states that The left-hand side is independent of
but unlike Szegő's original version, where
, Verblunsky's form does allow
[6] Subsequently, One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.
[7] Rakhmanov's theorem states that if the absolutely continuous part
is positive almost everywhere then the Verblunsky coefficients
The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.