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 ${\displaystyle F(0)=1}$.^[4] 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 ${\displaystyle f(z)}$ 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.^[4][5] The algorithm 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) via the recursion:^[6] $${\displaystyle f_{j+1}={\frac {1}{z}}{\frac {f_{j}(z)-\gamma _{j}}{1-{\overline {\gamma _{j}}}f_{j}(z)}},\quad f_{j}(0)\equiv \gamma _{j}\in \mathbb {D} ,}$$ which stops if ${\displaystyle f_{j}(z)\equiv e^{i\theta }=\gamma _{j}\in \mathbb {T} }$. One can invert the transformation as $${\displaystyle f(z)\equiv f_{0}(z)={\frac {\gamma _{0}+zf_{1}(z)}{1+{\overline {\gamma _{0}}}zf_{1}(z)}}}$$ or, equivalently, as continued fraction expansion of the Schur function $${\displaystyle f_{0}(z)=\gamma _{0}+{\frac {1-|\gamma _{0}|^{2}}{{\overline {\gamma _{0}}}+{\frac {1}{z\gamma _{1}+{\frac {z(1-|\gamma _{1}|^{2})}{{\overline {\gamma _{1}}}+{\frac {1}{z\gamma _{2}+\cdots }}}}}}}}}$$ by repeatedly using the fact that $${\displaystyle f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{j+1}(z)}}}}.}$$

• Orthogonal polynomials on the unit circle
• Szegő polynomial