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

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 Szegő recurrence

Szegő's recurrence states that

${\displaystyle \Phi _{0}(z)=1}$

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

where

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

is the polynomial with its coefficients reversed and complex conjugated, and where the Verblunsky coefficients ${\displaystyle \alpha _{n}}$ are complex numbers with absolute values less than 1.

Verblunsky's theorem

Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support.

Geronimus's theorem

Geronimus's theorem states that the Verblunsky coefficients of the measure μ are the Schur parameters of the function ${\displaystyle f}$ defined by the equations

${\displaystyle {\frac {1+zf(z)}{1-zf(z)}}=F(z)=\int {\frac {e^{i\theta }+z}{e^{i\theta }-z}}d\mu .}$

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.

Szegő's theorem

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 ${\displaystyle \mu }$.

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.

References

• 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, 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, 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3675-0, MR 2105089
• Szegő, Gábor (1920), "Beiträge zur Theorie der Toeplitzschen Formen" (PDF), Mathematische Zeitschrift, 6 (3–4): 167–202, doi:10.1007/BF01199955, ISSN 0025-5874
• Szegő, Gábor (1921), "Beiträge zur Theorie der Toeplitzschen Formen", Mathematische Zeitschrift, 9 (3–4): 167–190, doi:10.1007/BF01279027, ISSN 0025-5874
• Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications, XXIII, American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517