Schur class

In complex analysis, the Schur class is the set of holomorphic functions ${\displaystyle f(z)}$ defined on the open unit disk ${\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}$ and satisfying ${\displaystyle |f(z)|\leq 1}$ that solve the Schur problem: Given complex numbers ${\displaystyle c_{0},c_{1},\dotsc ,c_{n}}$, find a function $${\displaystyle f(z)=\sum _{j=0}^{n}c_{j}z^{j}+\sum _{j=n+1}^{\infty }f_{j}z^{j},}$$ which is analytic and bounded by 1 on the unit disk.¹ The method of solving this problem as well as similar problems (e.g. solving Toeplitz systems and Nevanlinna-Pick interpolation) is known as the Schur algorithm (also called coefficient stripping or layer stripping). One of the algorithm's most important properties is that it generates n + 1 orthogonal polynomials which can be used as orthonormal basis functions to expand any nth-order polynomial.² It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing.³

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.⁴⁵ 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 $${\displaystyle f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{j+1}(z)}}}}.}$$