- Source: Schur class
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}
that solve the Schur problem: 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
n
f
j
z
j
{\displaystyle 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
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:
f
j
+
1
=
1
z
f
j
(
z
)
−
γ
j
1
−
γ
j
¯
f
j
(
z
)
,
f
j
(
0
)
≡
γ
j
∈
D
,
{\displaystyle f_{j+1}={\frac {1}{z}}{\frac {f_{j}(z)-\gamma _{j}}{1-{\overline {\gamma _{j}}}f_{j}(z)}},\quad f_{j}(0)\equiv \gamma _{j}\in \mathbb {D} ,}
which stops if
f
j
(
z
)
≡
e
i
θ
=
γ
j
∈
T
{\displaystyle f_{j}(z)\equiv e^{i\theta }=\gamma _{j}\in \mathbb {T} }
. One can invert the transformation as
f
(
z
)
≡
f
0
(
z
)
=
γ
0
+
z
f
1
(
z
)
1
+
γ
0
¯
z
f
1
(
z
)
{\displaystyle f(z)\equiv f_{0}(z)={\frac {\gamma _{0}+zf_{1}(z)}{1+{\overline {\gamma _{0}}}zf_{1}(z)}}}
or, equivalently, as continued fraction expansion of the Schur function
f
0
(
z
)
=
γ
0
+
1
−
|
γ
0
|
2
γ
0
¯
+
1
z
γ
1
+
z
(
1
−
|
γ
1
|
2
)
γ
1
¯
+
1
z
γ
2
+
⋯
{\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 }}}}}}}}}
by repeatedly using the fact that
f
j
(
z
)
=
γ
j
+
1
−
|
γ
j
|
2
γ
j
¯
+
1
z
f
j
+
1
(
z
)
.
{\displaystyle f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{j+1}(z)}}}}.}
See also
Orthogonal polynomials on the unit circle
Szegő polynomial
References
Kata Kunci Pencarian:
- Jameela Jamil
- Mogok kerja SAG-AFTRA 2023
- Schur class
- Schur algorithm
- List of things named after Issai Schur
- Issai Schur
- Schur function
- Schur's lemma
- Hadamard product (matrices)
- A Man on the Inside
- Frobenius–Schur indicator
- Alternating group
