PlanetMath: inner function

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inner function

(Definition)

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

$\displaystyle f^*(e^{i\theta}) := \lim_{r\to 1, r<1} f(re^{i\theta}) .$

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

Definition 1 A bounded analytic function

is called an inner function if $\lvert f^*(e^{i\theta}) \rvert = 1$ almost everywhere. If

has no zeros on the unit disc, then

is called a singular inner function.

Theorem 1 Every inner function can be written as

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

where $\mu$ is a positive singular measure on $\partial \mathbb{D}$ , is a Blaschke product and $\lvert \alpha \rvert = 1$ is a constant.

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

Definition 2 Let

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

where

is a real valued Lebesgue integrable function on the unit circle and

is the Lebesgue measure. Then

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 $H^\infty$ functions.

Bibliography

1: John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.

"inner function" is owned by jirka.

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See Also: factorization theorem for $H^\infty$ functions

Also defines:

singular inner function, outer function

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Cross-references: holomorphic function, definitions, unit circle, Lebesgue integrable, real, function, Blaschke product, singular measure, positive, Lebesgue measure, almost everywhere, disc, bounded, limit, radial, unit disc, analytic function
There are 2 references to this entry.

This is version 3 of inner function, born on 2005-12-07, modified 2007-08-22.
Object id is 7522, canonical name is InnerFunction.
Accessed 3378 times total.

Classification:

AMS MSC:

30H05 (Functions of a complex variable :: Spaces and algebras of analytic functions)

Pending Errata and Addenda

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