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 .
Let
\mu
T=\{z\inC:|z|=1\}
\mu
d\mu=w(\theta)
d\theta | |
2\pi |
+d\mus
d\mus
d\theta/2\pi
w\inL1(T)
wd\theta/2\pi
d\mu
The orthogonal polynomials associated with
\mu
n | |
\Phi | |
n(z)=z |
+lowerorder
\int
j\Phi | |
\bar{z} | |
n(z)d\mu(z) |
=0, j=0,1,...,n-1
The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form
\Phin+1(z)=z\Phin(z)-\overline\alphan\Phi
*(z) | |
n |
\ast | |
\Phi | |
n+1 |
\ast | |
(z)=\Phi | |
n |
(z)-\alphanz\Phin(z)
n\geq0
\Phi0=1
*(z)=z | |
\Phi | |
n |
n\overline{\Phi | |
n(1/\overline{z})} |
\alphan
D=\{z\inC:|z|<1\}
\alphan=-\overline{\Phin+1(0)}
called the Verblunsky coefficients. Moreover,
\|\Phin+1\|2=
n | |
\prod | |
j=0 |
2) | |
(1-|\alpha | |
j| |
=
2 | |
(1-|\alpha | |
n\| |
d\mu
\alphan(d\mu)=\gamman
Verblunsky's theorem states that for any sequence of numbers
(0) | |
\{\alpha | |
j |
infty | |
\} | |
j=0 |
D
\mu
T
\alphaj(d\mu)=\alpha
(0) | |
j |
Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of
\mu
w
For any nontrivial probability measure
d\mu
T
2) | |
\prod | |
n| |
=\exp(
1 | |
2\pi |
2\pi | |
\int | |
0 |
logw(\theta)d\theta).
d\mus
d\mu=d\muac
d\mus ≠ 0
infty | |
\sum | |
n=0 |
2 | |
|\alpha | |
n| |
<infty \iff
1 | |
2\pi |
2\pi | |
\int | |
0 |
logw(\theta)d\theta>-infty
Rakhmanov's theorem states that if the absolutely continuous part
w
\mu
\alphan
The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.