Pseudocontinuations and the backward shift
Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)=\break (f(z)-f