# Beurling theorem

Let $f$ be a function in the Hardy class $H^2$ (cf. also Hardy classes). The vector space spanned by the functions $e^{in\theta}f$, $n\geq0$, is dense in $H^2$ if and only if $f$ is an outer function (cf. also Hardy classes).
This follows from the characterization of closed shift-invariant subspaces in $H^2$ as being of the form $gH^2$ with $g$ an inner function.