Blaschke product

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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

a0, a1, …

inside the unit disc.

Blaschke product, B(z), associated to 50 randomly chosen points in the unit disk.

ζ
=

e

2
π
i

/

3

{\displaystyle \zeta =e^{2\pi i/3}}

. B(z) is represented as a Matplotlib plot, using a version of the Domain coloring method.

Blaschke products were introduced by Wilhelm Blaschke (1915). They are related to Hardy spaces.

Contents

1 Definition
2 Szegő theorem
3 Finite Blaschke products
4 See also
5 References

Definition[edit]
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

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