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

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

###### Theorem.

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

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

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

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title factorization theorem for $H^{\infty}$ functions FactorizationTheoremForHinftyFunctions 2013-03-22 15:36:23 2013-03-22 15:36:23 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 30H05 InnerFunction