Vol. 43, No. 1, 1972

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ISSN: 0030-8730
A Phragmén-Lindelöf theorem with applications to (u,v) functions

Thomas Latimer Kriete, III and Marvin Rosenblum

Vol. 43 (1972), No. 1, 175–188
Abstract

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 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 (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 (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

    2     1−-|u(z)|2
|f(z)| ≦ K  1− |z|2  ,|z| > 1.

If u is inner of type C, it is proved that this necessary condition is also sufficient.

Mathematical Subject Classification
Primary: 30A42
Secondary: 30A78
Milestones
Received: 30 June 1971
Published: 1 October 1972
Authors
Thomas Latimer Kriete, III
Marvin Rosenblum