factorization theorem for $H^{\mathrm{\infty }}$ functions

Let $H^{\mathrm{\infty }}$ denote the bounded analytic functions on the unit disc.

Theorem.

Every $f\mathrm{\in }{H}^{\mathrm{\infty }}$ can be written as

$$f(z)=\alpha I(z)F(z)$$

where $\mathrm{|}\alpha \mathrm{|}\mathrm{=}\mathrm{1}$, $I$ is an inner function and $F$ is a bounded outer function. Conversely any function which can be so written is bounded.

Title	factorization theorem for $H^{\mathrm{\infty }}$ functions
Canonical name	FactorizationTheoremForHinftyFunctions
Date of creation	2013-03-22 15:36:23
Last modified on	2013-03-22 15:36:23
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	4
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30H05
Related topic	InnerFunction