In complex analysis, the Schur class is the set of holomorphic functions f ( z ) {\displaystyle f(z)} defined on the open unit disk D = { z ∈ C : | z | < 1 } {\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}} and satisfying | f ( z ) | ≤ 1 {\displaystyle |f(z)|\leq 1} .
Schur problem (or Carathéodory-Fejér interpolation) Given complex numbers c 0 , c 1 , … , c n {\displaystyle c_{0},c_{1},\dotsc ,c_{n}} , find a function
f ( z ) = ∑ j = 0 n c j z j + ∑ j = n + 1 ∞ f j z j , {\displaystyle f(z)=\sum _{j=0}^{n}c_{j}z^{j}+\sum _{j=n+1}^{\infty }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 μ {\displaystyle d\mu } on the unit circle T = { z ∈ C : | z | = 1 } {\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}} given by
F ( z ) = ∫ e i θ + z e i θ − z d μ ( θ ) , {\displaystyle F(z)=\int {\frac {e^{i\theta }+z}{e^{i\theta }-z}}d\mu (\theta ),}
where ∫ d μ ( θ ) = 1 {\displaystyle \int d\mu (\theta )=1} implies F ( 0 ) = 1 {\displaystyle F(0)=1} . Then the association
F ( z ) = 1 + z f ( z ) 1 − z f ( z ) , {\displaystyle F(z)={\frac {1+zf(z)}{1-zf(z)}},}
sets up a one-to-one correspondence between Carathéodory functions and Schur functions f ( z ) {\displaystyle f(z)} given by the inverse formula:
f ( z ) = z − 1 ( F ( z ) − 1 F ( z ) + 1 ) . {\displaystyle 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 ≡ f 0 , f 1 , … , f n , … {\displaystyle f\equiv f_{0},f_{1},\dotsc ,f_{n},\dotsc } and Schur parameters γ 0 , γ 1 , … , γ n , … {\displaystyle \gamma _{0},\gamma _{1},\dotsc ,\gamma _{n},\dotsc } (also called Verblunsky coefficient or reflection coefficient) via the recursion:
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