Orthogonal polynomials on the unit circle

Definition

Let \mu be a probability measure on the unit circle \mathbb{T} ={z\in\mathbb{C} :|z|=1} and assume \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 :d\mu =w(\theta) \frac{d\theta}{2\pi} + d\mu_s, where d\mu_s is * singular* with respect to d\theta/2\pi and w \in L^{1}(\mathbb{T}) with wd\theta/2\pi the absolutely continuous part of d\mu.

The orthogonal polynomials associated with \mu are defined as :\Phi_n(z)=z^n + \text{lower order}, such that :\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 :\Phi_{n+1}(z)=z\Phi_n(z)-\overline\alpha_n\Phi_n^(z) :\Phi_{n+1}^{\ast}(z)=\Phi_n^{\ast}(z)-\alpha_n z\Phi_n(z) for n \geq 0 and initial condition \Phi_0=1, with :\Phi_n^(z)=z^n\overline{\Phi_n(1/\overline{z})} and constants \alpha_n in the open unit disk \mathbb{D} = { z\in \mathbb{C} : |z| given by :\alpha_n = -\overline{\Phi_{n+1}(0)}

called the Verblunsky coefficients. Moreover, :|\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 d\mu are the Schur parameters: : \alpha_n(d\mu) = \gamma_n

Verblunsky's theorem

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

Baxter's theorem

Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of \mu form an absolutely convergent series and the weight function w is strictly positive everywhere.

Szegő's theorem

For any nontrivial probability measure d\mu on \mathbb{T}, Verblunsky's form of Szegő's theorem states that :\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 d\mu_s but unlike Szegő's original version, where d\mu = d\mu_{ac}, Verblunsky's form does allow d\mu_s \neq 0 . Subsequently, :\sum_{n = 0}^\infty |\alpha_n|^2 -\infty. One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.

Rakhmanov's theorem

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

Examples

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

Notes

References