schur class

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

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

where $\backslash int\; d\backslash mu(\backslash theta)\; =\; 1$ implies $F(0)=1$.{{Citation | last1=Simon | first1=Barry | author1-link=Barry Simon | title=Orthogonal polynomials on the unit circle. Part 1. Classical theory | url=https://books.google.com/books?id=d94r7kOSnKcC | publisher=American Mathematical Society | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-3446-6 | mr=2105088 | year=2005 | volume=54}} Then the association

:$F(z)\; =\; \backslash frac\{1+zf(z)\}\{1-zf(z)\}$

sets up a one-to-one correspondence between Carathéodory functions and Schur functions $f(z)$ given by the inverse formula:

:$f(z)\; =\; z^\{-1\}\backslash left(\; \backslash frac\{F(z)-1\}\{F(z)+1\}\; \backslash right)$

Schur's algorithm is an iterative construction based on Möbius transformations that maps one Schur function to another.{{Cite book |last=Conway |first=John B. |title=Functions of One Complex Variable I (Graduate Texts in Mathematics 11) |title-link=Graduate Texts in Mathematics |publisher=Springer-Verlag |year=1978 |isbn=978-0-387-90328-6|page=127}} The algorithm defines an infinite sequence of Schur functions $f\backslash equiv\; f\_0,f\_1,\backslash dotsc,f\_n,\backslash dotsc$ and Schur parameters $\backslash gamma\_0,\backslash gamma\_1,\backslash dotsc,\backslash gamma\_n,\backslash dotsc$ (also called Verblunsky coefficient or reflection coefficient) via the recursion:{{Citation | last =Simon| first =Barry | author-link=Barry Simon | title=Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials| url=https://books.google.com/books?id=e9R5gcz_x0YC | date=2010|publisher=Princeton University Press|isbn=978-0-691-14704-8}}

:$f\_\{j+1\}=\backslash frac\{1\}\{z\}\backslash frac\{f\_j(z)-\backslash gamma\_j\}\{1-\backslash overline\{\backslash gamma\_j\}f\_j(z)\},\; \backslash quad\; f\_j(0)\backslash equiv\; \backslash gamma\_j\; \backslash in\; \backslash mathbb\{D\},$

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

:$f(z)\backslash equiv\; f\_0\; (z)\; =\; \backslash frac\{\backslash gamma\_0\; +\; zf\_1(z)\}\{1\; +\; \backslash overline\{\backslash gamma\_0\}\; z\; f\_1(z)\; \}$

or, equivalently, as continued fraction expansion of the Schur function

:$f\_0(z)=\backslash gamma\_0+\backslash frac\{1-|\backslash gamma\_0|^2\}\{\backslash overline\; \{\backslash gamma\_0\}+\backslash frac\{1\}\{z\; \backslash gamma\_1+\backslash frac\{z(1-|\backslash gamma\_1|^2)\}\{\backslash overline\; \{\backslash gamma\_1\}+\backslash frac\{1\}\{z\backslash gamma\_2+\backslash cdots\}\}\}\}$

by repeatedly using the fact that

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

• Orthogonal polynomials on the unit circle
• Szegő polynomial

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Category:Complex analysis