Schur class

In complex analysis, the Schur class is the set of holomorphic functions $$ f(z)$$ defined on the open unit disk $$ \mathbb{D} = \{ z\in \mathbb{C} : |z|<1\} $$ and satisfying $$ |f(z)| \leq 1$$ that solve the Schur problem: Given complex numbers $$c_0,c_1,\dotsc,c_n$$, find a function
 * $$f(z) = \sum_{j=0}^{n} c_j z^j + \sum_{j=n+1}^{n}f_j z^j$$

which is analytic and bounded by $1$ on the unit disk. 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 $n$th-order polynomial. It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing.

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


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

where $$\int d\mu(\theta) = 1$$ implies $$ F(0)=1$$. Then the association
 * $$ F(z) = \frac{1+zf(z)}{1-zf(z)}$$

sets up a one-to-one correspondence between Carathéodory functions and Schur functions $$ f(z)$$ given by the inverse formula:
 * $$ f(z) = z^{-1}\left( \frac{F(z)-1}{F(z)+1} \right)$$

Schur algorithm
Schur's algorithm is an iterative construction based on Möbius transformations that maps one Schur function to another. The algorithm defines an infinite sequence of Schur functions $$f\equiv f_0,f_1,\dotsc,f_n,\dotsc$$ and Schur parameters $$ \gamma_0,\gamma_1,\dotsc,\gamma_n,\dotsc$$ (also called Verblunsky coefficient or reflection coefficient) via the recursion:
 * $$f_{j+1}=\frac{1}{z}\frac{f_j(z)-\gamma_j}{1-\overline{\gamma_j}f_j(z)}, \quad f_j(0)\equiv \gamma_j \in \mathbb{D},$$

which stops if $$f_j(z)\equiv e^{i\theta} = \gamma_j \in \mathbb{T} $$. One can invert the transformation as
 * $$ f(z)\equiv f_0 (z) = \frac{\gamma_0 + zf_1(z)}{1 + \overline{\gamma_0} z f_1(z) }$$

or, equivalently, as continued fraction expansion of the Schur function
 * $$ f_0(z)=\gamma_0+\frac{1-|\gamma_0|^2}{\overline {\gamma_0}+\frac{1}{z \gamma_1+\frac{z(1-|\gamma_1|^2)}{\overline {\gamma_1}+\frac{1}{z\gamma_2+\cdots}}}}$$

by repeatedly using the fact that
 * $$ f_j(z)=\gamma_j+\frac{1-|\gamma_j|^2}{\overline {\gamma_j}+\frac{1}{zf_{j+1}(z)}}.$$