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 calledan 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).