Orthogonal polynomials on the unit circle

Suppose that ${\displaystyle \mu }$ is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to ${\displaystyle \mu }$ are the polynomials ${\displaystyle \Phi _{n}(z)}$ with leading term ${\displaystyle z^{n}}$ that are orthogonal with respect to the measure ${\displaystyle \mu }$.

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

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

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 ${\displaystyle \alpha _{n}\in \mathbb {C} }$ given by

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

called the Verblunsky coefficients.[1]^: 44 Subsequently, Geronimus' theorem states that the Verblunsky coefficients associated with ${\displaystyle d\mu }$ are the Schur parameters:[1]^: 74

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

Verblunsky's form of Szegő's theorem states that

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

where ${\displaystyle wd\theta /2\pi }$ is the absolutely continuous part of the measure ${\displaystyle d\mu (\theta )=wd\theta /2\pi +d\mu _{s}}$. Verblunsky's form also allows for a non-zero singular part while ${\displaystyle d\mu _{s}=0}$ in Szegő's original version.[1]^: 29

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.

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

• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials on the unit circle", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, MR 2105088
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 2. Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3675-0, MR 2105089
• Szegő, Gábor (1920), "Beiträge zur Theorie der Toeplitzschen Formen", Mathematische Zeitschrift, 6 (3–4): 167–202, doi:10.1007/BF01199955, ISSN 0025-5874, S2CID 118147030
• Szegő, Gábor (1921), "Beiträge zur Theorie der Toeplitzschen Formen", Mathematische Zeitschrift, 9 (3–4): 167–190, doi:10.1007/BF01279027, ISSN 0025-5874, S2CID 125157848
• Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications, vol. XXIII, American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517