inner function

If $f:𝔻\to ℂ$ is an analytic function on the unit disc, we denote by $f^{*}(e^{i\theta })$ the radial limit of $f$ where it exists, that is

A bounded analytic function on the disc will have radial limits almost everywhere (with respect to the Lebesgue measure on the $\partial 𝔻$).

Definition.

A bounded analytic function $f$ is called an inner function if $|f^{*}(e^{i\theta })|=1$ almost everywhere. If $f$ has no zeros on the unit disc, then $f$ is called a singular inner function.

Theorem.

Every inner function can be written as

$$f(z):=\alpha B(z)\exp \left(-\int \frac{e^{i\theta }+z}{e^{i\theta }-z}𝑑\mu (e^{i\theta })\right),$$

where $\mu$ is a positive singular measure on $\mathrm{\partial }\mathit{}\mathrm{D}$, $B\mathit{}\mathrm{(}z\mathrm{)}$ is a Blaschke product and $\mathrm{|}\alpha \mathrm{|}\mathrm{=}\mathrm{1}$ is a constant.

Note that all the zeros of the function come from the Blaschke product.

Definition.

Let

$$f(z):=\exp \left(\int \frac{e^{i\theta }+z}{e^{i\theta }-z}h(e^{i\theta })𝑑m(e^{i\theta })\right),$$

where $h$ is a real valued Lebesgue integrable function on the unit circle and $m$ is the Lebesgue measure. Then $f$ is called an outer function.

The significance of these definitions is that every bounded holomorphic function can be written as an inner function times an outer function. See the factorization theorem for $H^{\mathrm{\infty }}$ functions (http://planetmath.org/FactorizationTheoremForHinftyFunctions).

Title	inner function
Canonical name	InnerFunction
Date of creation	2013-03-22 15:36:20
Last modified on	2013-03-22 15:36:20
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 30H05
Related topic	FactorizationTheoremForHinftyFunctions
Defines	singular inner function
Defines	outer function