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 Szegő (1920, 1921, 1939).

Definition

Let ${\displaystyle \mu }$ be a probability measure on the unit circle ${\displaystyle \mathbb {T} =\{z\in \mathbb {C}$ :|z|=1\}} and assume ${\displaystyle \mu }$ 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

${\displaystyle d\mu =w(\theta ){\frac {d\theta }{2\pi }}+d\mu _{s}}$,

where ${\displaystyle d\mu _{s}}$ is singular with respect to ${\displaystyle d\theta /2\pi }$ and ${\displaystyle w\in L^{1}(\mathbb {T} )}$ with ${\displaystyle wd\theta /2\pi }$ the absolutely continuous part of ${\displaystyle d\mu }$.^[1]

The orthogonal polynomials associated with ${\displaystyle \mu }$ are defined as

${\displaystyle \Phi _{n}(z)=z^{n}+{\text{lower order}}}$,

such that

${\displaystyle \int {\bar {z}}^{j}\Phi _{n}(z)\,d\mu (z)=0,\quad j=0,1,\dots ,n-1}$.

The Szegő recurrence

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

${\displaystyle \Phi _{n+1}(z)=z\Phi _{n}(z)-{\overline {\alpha }}_{n}\Phi _{n}^{*}(z)}$

${\displaystyle \Phi _{n+1}^{\ast }(z)=\Phi _{n}^{\ast }(z)-\alpha _{n}z\Phi _{n}(z)}$

for ${\displaystyle n\geq 0}$ and initial condition ${\displaystyle \Phi _{0}=1}$, with

${\displaystyle \Phi _{n}^{*}(z)=z^{n}{\overline {\Phi _{n}(1/{\overline {z}})}}}$

and constants ${\displaystyle \alpha _{n}}$ in the open unit disk ${\displaystyle \mathbb {D} =\{z\in \mathbb {C}$ :|z|<1\}} given by

${\displaystyle \alpha _{n}=-{\overline {\Phi _{n+1}(0)}}}$

called the Verblunsky coefficients. ^[2] Moreover,

${\displaystyle \|\Phi _{n+1}\|^{2}=\prod _{j=0}^{n}(1-|\alpha _{j}|^{2})=(1-|\alpha _{n}|^{2})\|\Phi _{n}\|^{2}}$.

Geronimus' theorem states that the Verblunsky coefficients associated with ${\displaystyle d\mu }$ are the Schur parameters:^[3]

${\displaystyle \alpha _{n}(d\mu )=\gamma _{n}}$

Verblunsky's theorem

Verblunsky's theorem states that for any sequence of numbers ${\displaystyle \{\alpha _{j}^{(0)}\}_{j=0}^{\infty }}$ in ${\displaystyle \mathbb {D} }$ there is a unique nontrivial probability measure ${\displaystyle \mu }$ on ${\displaystyle \mathbb {T} }$ with ${\displaystyle \alpha _{j}(d\mu )=\alpha _{j}^{(0)}}$.^[4]

Baxter's theorem

Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of ${\displaystyle \mu }$ form an absolutely convergent series and the weight function ${\displaystyle w}$ is strictly positive everywhere.^[5]

Szegő's theorem

For any nontrivial probability measure ${\displaystyle d\mu }$ on ${\displaystyle \mathbb {T} }$, Verblunsky's form of Szegő's theorem states that

${\displaystyle \prod _{n=0}^{\infty }(1-|\alpha _{n}|^{2})=\exp {\big (}{\frac {1}{2\pi }}\int _{0}^{2\pi }\log w(\theta )\,d\theta {\big )}.}$

The left-hand side is independent of ${\displaystyle d\mu _{s}}$ but unlike Szegő's original version, where ${\displaystyle d\mu =d\mu _{ac}}$, Verblunsky's form does allow ${\displaystyle d\mu _{s}\neq 0}$.^[6] Subsequently,

${\displaystyle \sum _{n=0}^{\infty }|\alpha _{n}|^{2}<\infty \;\iff \;{\frac {1}{2\pi }}\int _{0}^{2\pi }\log w(\theta )\,d\theta >-\infty }$.

One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.^[7]

Rakhmanov's theorem

Rakhmanov's theorem states that if the absolutely continuous part ${\displaystyle w}$ of the measure ${\displaystyle \mu }$ is positive almost everywhere then the Verblunsky coefficients ${\displaystyle \alpha _{n}}$ tend to 0.

Examples

The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.

See also

• Trigonometric moment problem
• Schur class