- Source: Blaschke product
In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers
a
0
,
a
1
,
…
{\displaystyle a_{0},\ a_{1},\ldots }
inside the unit disc, with the property that the magnitude of the function is constant along the boundary of the disc.
Blaschke products were introduced by Wilhelm Blaschke (1915). They are related to Hardy spaces.
Definition
A sequence of points
(
a
n
)
{\displaystyle (a_{n})}
inside the unit disk is said to satisfy the Blaschke condition when
∑
n
(
1
−
|
a
n
|
)
<
∞
.
{\displaystyle \sum _{n}(1-|a_{n}|)<\infty .}
Given a sequence obeying the Blaschke condition, the Blaschke product is defined as
B
(
z
)
=
∏
n
B
(
a
n
,
z
)
{\displaystyle B(z)=\prod _{n}B(a_{n},z)}
with factors
B
(
a
,
z
)
=
|
a
|
a
a
−
z
1
−
a
¯
z
{\displaystyle B(a,z)={\frac {|a|}{a}}\;{\frac {a-z}{1-{\overline {a}}z}}}
provided
a
≠
0
{\displaystyle a\neq 0}
. Here
a
¯
{\displaystyle {\overline {a}}}
is the complex conjugate of
a
{\displaystyle a}
. When
a
=
0
{\displaystyle a=0}
take
B
(
0
,
z
)
=
z
{\displaystyle B(0,z)=z}
.
The Blaschke product
B
(
z
)
{\displaystyle B(z)}
defines a function analytic in the open unit disc, and zero exactly at the
a
n
{\displaystyle a_{n}}
(with multiplicity counted): furthermore it is in the Hardy class
H
∞
{\displaystyle H^{\infty }}
.
The sequence of
a
n
{\displaystyle a_{n}}
satisfying the convergence criterion above is sometimes called a Blaschke sequence.
Szegő theorem
A theorem of Gábor Szegő states that if
f
∈
H
1
{\displaystyle f\in H^{1}}
, the Hardy space with integrable norm, and if
f
{\displaystyle f}
is not identically zero, then the zeroes of
f
{\displaystyle f}
(certainly countable in number) satisfy the Blaschke condition.
Finite Blaschke products
Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that
f
{\displaystyle f}
is an analytic function on the open unit disc such
that
f
{\displaystyle f}
can be extended to a continuous function on the closed unit disc
Δ
¯
=
{
z
∈
C
∣
|
z
|
≤
1
}
{\displaystyle {\overline {\Delta }}=\{z\in \mathbb {C} \mid |z|\leq 1\}}
that maps the unit circle to itself. Then
f
{\displaystyle f}
is equal to a finite Blaschke product
B
(
z
)
=
ζ
∏
i
=
1
n
(
z
−
a
i
1
−
a
i
¯
z
)
m
i
{\displaystyle B(z)=\zeta \prod _{i=1}^{n}\left({{z-a_{i}} \over {1-{\overline {a_{i}}}z}}\right)^{m_{i}}}
where
ζ
{\displaystyle \zeta }
lies on the unit circle and
m
i
{\displaystyle m_{i}}
is the multiplicity of the zero
a
i
{\displaystyle a_{i}}
,
|
a
i
|
<
1
{\displaystyle |a_{i}|<1}
. In particular, if
f
{\displaystyle f}
satisfies the condition above and has no zeros inside the unit circle, then
f
{\displaystyle f}
is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function
log
(
|
f
(
z
)
|
)
{\displaystyle \log(|f(z)|)}
.
See also
Hardy space
Weierstrass product
Complex Analysis
References
Blaschke, W. (1915). "Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen". Berichte Math.-Phys. Kl. (in German). 67. Sächs. Gesell. der Wiss. Leipzig: 194–200.
Colwell, Peter (1985). Blaschke Products. Ann Arbor, Michigan: University of Michigan Press. ISBN 0-472-10065-3. MR 0779463.
Conway, John B. (1996). Functions of a Complex Variable II. Graduate Texts in Mathematics. Vol. 159. Springer-Verlag. pp. 273–274. ISBN 0-387-94460-5.
Tamrazov, P.M. (2001) [1994]. "Blaschke product". Encyclopedia of Mathematics. EMS Press.
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