Significance and motivation for outer functions

Question

Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The Hardy spaces on $\mathbb{D}$ are defined as: $$H^{p}:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup _{r < 1}\int ^{2\pi }_{0}\left| f\left( re^{i\theta}\right) \right| ^{p}d\theta < \infty \right\} \;\;\;\;(0<p<\infty), $$ $$H^{\infty }:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup_{z\in D}\left| f\left( z\right) \right| < \infty \right\}.$$ A function $g\in H^p(\mathbb{D})$ is outer if there exists a function $G:\mathbb{T}\longrightarrow [0,\infty)$ with $G\in L^1(\mathbb{T})$ such that $$g\left( z\right) =\alpha \text{exp}\left( \int ^{2\pi }_{0}\dfrac {e^{i\theta }+z}{e^{i\theta }-z}G\left( e^{i\theta }\right) \dfrac {d\theta }{2\pi }\right) \qquad(z\in \mathbb{D})$$ and $|\alpha|=1$.

The definition of a outer function seems so involved. Can anyone tell what led to defining outer functions as such? What would be the motivation behind remembering such a definition?

I know that outer functions in a Hardy space are important, for example if we consider the canonical factorization of an element of a Hardy space. Can anyone mention some other major utility of outer functions?

Answer 1

Outer functions are important first of all in connection with the so-called inner-outer factorization. This is the factorization of an arbitrary bounded function $f$ as: $$f=BGH,$$ where $B$ is a Blaschke product, $G$ is an inner function and $H$ is outer. So the inner function stands inside, surrounded by $B$ and $H$, but $B$ already had a name when this was introduced:-) Many authors call the product $BG$ an inner function, not separating the Blaschke product. So they were definitely introduced together with inner functions, and this partially justifies the terminology. I believe that this was by Beurling in his seminal paper on the invariant subspaces of the shift operator. However the representation itself is much older, it is also called Herglotz representation. The paper by Beurling shows the importance of such factorization. Of course inner functions are at least as important as outer ones.