Schur class

In mathematics, the Schur class consists of the Schur functions: the holomorphic functions from the open unit disk to the closed unit disk. These functions were studied by Schur (1918).

The Schur parameters γ_j of a Schur function f₀ are defined recursively by

${\displaystyle \gamma _{j}=f_{j}(0)}$

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

The Schur parameters γ_j all have absolute value at most 1.

This gives a continued fraction expansion of the Schur function f₀ 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

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