A well-known theorem of
Paley and Wiener asserts that if f is an entire function, its restriction to the real line
belongs to the Hilbert space ℱ∗ L2(−τ,τ) (where ℱ is the Fourier-Plancherel
operator) if and only if f is square integrable on the real axis and satisfies
|f(z)|≦ Keτ|Imz| for some positive K. The “if” part of this result may be viewed as a
Phragmén-Lindelöf type theorem. The pair (eiτx,eiτx) of inner functions can be
associated with the above mentioned Hilbert space in a natural way. By replacing
this pair by a more general pair (u,v) of inner functions it is possible to define a
space ℳ(u,v) of analytic functions similar to the Paley-Wiener space. For a certain
class of inner functions (those of “type C”) it is shown that membership in
ℳ(u,v) is implied by an inequality analogous to the exponential inequality
above.
A second application of our results is to star-invariant subspaces of the Hardy
space H2. It is well known that if u is an inner function on the circle and f is in H2,
then in order for f to be in (uH2)⊥ it is necessary for f to have a meromorphic
pseudocontinuation to |z| > 1 satisfying
If u is inner of type C, it is proved that this necessary condition is also
sufficient.
