Blaschke Product - Finite Blaschke Products

Finite Blaschke Products

Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that f is an analytic function on the open unit disc such that f can be extended to a continuous function on the closed unit disc

which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product

B(z)=\zeta\prod_{i=1}^n\left({{z-a_i}\over {1-\overline{a_i}z}}\right)^{m_i}

where ζ lies on the unit circle and mi is the multiplicity of the zero ai, |ai| < 1. In particular, if ƒ satisfies the condition above and has no zeros inside the unit circle then ƒ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|ƒ(z)|)).

Read more about this topic:  Blaschke Product

Famous quotes containing the words finite and/or products:

    “Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are “prefabricated” in the sense that we don’t coin new ones every time we speak.”
    —David Lodge (b. 1935)

    “The reality is that zero defects in products plus zero pollution plus zero risk on the job is equivalent to maximum growth of government plus zero economic growth plus runaway inflation.”
    —Dixie Lee Ray (b. 1924)