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, 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.

A sequence of points

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

is the complex conjugate of

defines a function analytic in the open unit disc, and zero exactly at the

(with multiplicity counted): furthermore it is in the Hardy class

satisfying the convergence criterion above is sometimes called a Blaschke sequence.

A theorem of Gábor Szegő states that if

, the Hardy space with integrable norm, and if

(certainly countable in number) satisfy the Blaschke condition.

Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that

is an analytic function on the open unit disc such that

can be extended to a continuous function on the closed unit disc that maps the unit circle to itself.

is equal to a finite Blaschke product where

lies on the unit circle and

satisfies the condition above and has no zeros inside the unit circle, then

is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function