inside the unit disc, with the property that the magnitude of the function is constant along the boundary of the disc.
Файл:Blaschke-product.pngBlaschke product, <math>B(z)</math>, associated to 50 randomly chosen points in the unit disk. B(z) is represented as a Matplotlib plot, using a version of the Domain coloring method.
provided <math>a\neq 0</math>. Here <math>\overline{a}</math> is the complex conjugate of <math>a</math>. When <math>a=0</math> take <math>B(0,z)=z</math>.
The Blaschke product <math>B(z)</math> defines a function analytic in the open unit disc, and zero exactly at the <math>a_n</math> (with multiplicity counted): furthermore it is in the Hardy class <math>H^\infty</math>.[1]
The sequence of <math>a_n</math> satisfying the convergence criterion above is sometimes called a Blaschke sequence.
Szegő theorem
A theorem of Gábor Szegő states that if <math>f\in H^1</math>, the Hardy space with integrable norm, and if <math>f</math> is not identically zero, then the zeroes of <math>f</math> (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 <math>f</math> is an analytic function on the open unit disc such
that <math>f</math> can be extended to a continuous function on the closed unit disc
where <math>\zeta</math> lies on the unit circle and <math>m_i</math> is the multiplicity of the zero <math>a_i</math>,
<math>|a_i|<1</math>. In particular, if <math>f</math> satisfies the condition above and has no zeros inside the unit circle, then <math>f</math> is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function <math>\log(|f(z)|)</math>.
