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.

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

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

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

The functions f_j are also Schur functions as 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

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

which gives

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

References

• Schur, I. ({1918},), "Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. I, II", J. Reine Angew. Math., (in German), 147, , Berlin: Walter de Gruyter: 205--232}, doi:10.1515/crll.1917.147.205, Zbl 46.0475.01 {{citation}}: Check |zbl= value (help); Check date values in: |year= (help)CS1 maint: extra punctuation (link) CS1 maint: year (link)
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, MR 2105088
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 2. Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3675-0, MR 2105089