Orthogonal polynomials on the unit circle

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

:$d\backslash mu\; =w(\backslash theta)\; \backslash frac\{d\backslash theta\}\{2\backslash pi\}\; +\; d\backslash mu\_s$,

where $d\backslash mu\_s$ is singular with respect to $d\backslash theta/2\backslash pi$ and $w\; \backslash in\; L^\{1\}(\backslash mathbb\{T\})$ with $wd\backslash theta/2\backslash pi$ the absolutely continuous part of $d\backslash mu$.{{sfn|Simon|2005a|page=43}}

The orthogonal polynomials associated with $\backslash mu$ are defined as

:$\backslash Phi\_n(z)=z^n\; +\; \backslash text\{lower\; order\}$,

such that

:$\backslash int\; \backslash bar\{z\}^j\backslash Phi\_n(z)\backslash ,d\backslash mu(z)\; =0,\; \backslash quad\; j\; =\; 0,1,\backslash dots,n-1$.

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

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

:$\backslash Phi\_\{n+1\}^\{\backslash ast\}(z)=\backslash Phi\_n^\{\backslash ast\}(z)-\backslash alpha\_n\; z\backslash Phi\_n(z)$

for $n\; \backslash geq\; 0$ and initial condition $\backslash Phi\_0=1$, with

:$\backslash Phi\_n^*(z)=z^n\backslash overline\{\backslash Phi\_n(1/\backslash overline\{z\})\}$

and constants $\backslash alpha\_n$ in the open unit disk given by

:$\backslash alpha\_n\; =\; -\backslash overline\{\backslash Phi\_\{n+1\}(0)\}$

called the Verblunsky coefficients. {{sfn|Simon|2010|page=44}} Moreover,

:$\backslash |\backslash Phi\_\{n+1\}\backslash |^\{2\}\; =\; \backslash prod\_\{j=0\}^\{n\}\; (1-|\backslash alpha\_j|^2)\; =\; (1-|\backslash alpha\_n|^2)\backslash |\backslash Phi\_n\backslash |^2$.

Geronimus' theorem states that the Verblunsky coefficients associated with $d\backslash mu$ are the Schur parameters:{{sfn|Simon|2010|page=74}}

:$\backslash alpha\_n(d\backslash mu)\; =\; \backslash gamma\_n$

Verblunsky's theorem states that for any sequence of numbers $\backslash \{\backslash alpha\_\{j\}^\{(0)\}\backslash \}\_\{j=0\}^\{\backslash infty\}$ in $\backslash mathbb\{D\}$ there is a unique nontrivial probability measure $\backslash mu$ on $\backslash mathbb\{T\}$ with $\backslash alpha\_j(d\backslash mu)=\backslash alpha\_j^\{(0)\}$.{{sfn|Schmüdgen|2017|page=265}}

Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of $\backslash mu$ form an absolutely convergent series and the weight function $w$ is strictly positive everywhere.{{sfn|Simon|2005a|page=313}}

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

:$\backslash prod\_\{n\; =\; 0\}^\backslash infty(1-|\backslash alpha\_n|^2)\; =\; \backslash exp\backslash big(\backslash frac\{1\}\{2\backslash pi\}\backslash int\_0^\{2\backslash pi\}\backslash log\; w(\backslash theta)\backslash ,d\backslash theta\backslash big).$

The left-hand side is independent of $d\backslash mu\_s$ but unlike Szegő's original version, where $d\backslash mu\; =\; d\backslash mu\_\{ac\}$, Verblunsky's form does allow $d\backslash mu\_s\; \backslash neq\; 0$.{{sfn|Simon|2010|page=29}} Subsequently,

:.

One of the consequences is the existence of a mixed spectrum for discretized Schrödinger operators.{{sfn | Totik | 2016 | page=269}}

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

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

• Trigonometric moment problem
• Schur class

{{Reflist}}

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• {{cite book | last=Schmüdgen | first=Konrad | series= Graduate Texts in Mathematics|title=The Moment Problem | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=277 | isbn=978-3-319-64545-2 | issn=0072-5285 | doi=10.1007/978-3-319-64546-9}}
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Category:Orthogonal polynomials