Schur class

Consider the Carathéodory function of a unique probability measure ${\displaystyle d\mu }$ on the unit circle ${\displaystyle \mathbb {T} =\{z\in \mathbb {C}$ :|z|=1\}} given by

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

where ${\displaystyle \int d\mu (\theta )=1}$ implies ${\displaystyle F(0)=1}$.[1] Then the association

${\displaystyle F(z)={\frac {1+zf(z)}{1-zf(z)}}}$

sets up a one-to-one correspondence between Carathéodory functions and Schur functions given by the inverse formula:

${\displaystyle f(z)=z^{-1}\left({\frac {F(z)-1}{F(z)+1}}\right)}$

Schur's algorithm is an iterative construction based on Möbius transformations that maps one Schur function to another.[1][2] The algorithm, written as

${\displaystyle \gamma _{j}=f_{j}(0),\quad |\gamma _{j}|<1}$

${\displaystyle f_{j+1}={\frac {1}{z}}{\frac {f_{j}(z)-\gamma _{j}}{1-{\overline {\gamma _{j}}}f_{j}(z)}}}$

defines an infinite sequence of Schur functions ${\displaystyle f\equiv f_{0},f_{1},\dotsc ,f_{n},\dotsc }$ and Schur parameters ${\displaystyle \gamma _{0},\gamma _{1},\dotsc ,\gamma _{n},\dotsc }$ (also called Verblunsky coefficient or reflection coefficient).[3] The algorithm stops in case ${\displaystyle f_{j}(z)\equiv e^{i\theta }=\gamma _{j}}$.

One can invert the transformation by repeatedly using the fact that

${\displaystyle f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{j+1}(z)}}}}}$

which gives a continued fraction expansion of the Schur function

${\displaystyle f(z)\equiv f_{0}(z)=\gamma _{0}+{\frac {1-|\gamma _{0}|^{2}}{{\overline {\gamma _{0}}}+{\frac {1}{z\gamma _{1}+{\frac {z(1-|\gamma _{1}|^{2})}{{\overline {\gamma _{1}}}+{\frac {1}{z\gamma _{2}+\cdots }}}}}}}}}$

and in particular

${\displaystyle f(z)={\frac {\gamma _{0}+zf_{1}(z)}{1+{\overline {\gamma _{0}}}zf_{1}(z)}}}$.

• Orthogonal polynomials on the unit circle
• Szegő polynomial

• Schur, I. (1918), "Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. I, II", J. Reine Angew. Math. (in German), Berlin: Walter de Gruyter, 147: 205–232, doi:10.1515/crll.1917.147.205, JFM 46.0475.01
• 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