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 $|f(z)|\le 1$ that solve the Schur problem: Given complex numbers $c_0,c_1,\dots ,c_n$, 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 nth-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 $f\equiv f_0,f_1,\dots ,f_n,\dots$ and Schur parameters ${\gamma }_0,{\gamma }_1,\dots ,{\gamma }_n,\dots$ (also called Verblunsky coefficient or reflection coefficient) via the recursion:

$f_{j+1}=\frac{1}{z}\frac{f_j(z)-{\gamma }_j}{1-\overline{{\gamma }_j}{f}_j(z)},f_j(0)\equiv {\gamma }_j\in \mathbb{𝔻},$

which stops if $f_j(z)\equiv e^{i\theta }={\gamma }_j\in \mathbb{𝕋}$. One can invert the transformation as

$f(z)\equiv f_0(z)=\frac{{\gamma }_0+z{f}_1(z)}{1+\overline{{\gamma }_0}z{f}_1(z)}$

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

$f_j(z)={\gamma }_j+\frac{1-|{\gamma }_j{|}^2}{\overline{{\gamma }_j}+\frac{1}{z{f}_{j+1}(z)}}.$

See also

• Orthogonal polynomials on the unit circle
• Szegő polynomial