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

inside the unit disc.

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

A sequence of points ${\displaystyle (a_{n})}$ inside the unit disk is said to satisfy the Blaschke condition when

Given a sequence obeying the Blaschke condition, the Blaschke product is defined as

with factors

provided a ≠ 0. Here ${\displaystyle {\overline {a}}}$ is the complex conjugate of a. When a = 0 take B(0,z) = z.

The Blaschke product B(z) defines a function analytic in the open unit disc, and zero exactly at the a_n (with multiplicity counted): furthermore it is in the Hardy class ${\displaystyle H^{\infty }}$.

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