On the structure of analytic vectors for the Schrödinger representation

Abstract

This article deals with the structure of analytic and entire vectors for the Schrödinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.

Variations on a theorem of Beurling

“…where C 2 = C 1 e 1/2 . Similar arguments can be given to verify that Bf (z)B f (−z) also satisfies the estimate of (6). Now Since K a (f ) = K a ( f ), the same estimate (6) holds when f is replaced by f and then in view of B f (z) = Bf (−iz), we have…”

“…The theorem is proved by estimating the Taylor coefficients of the Bargmann transform of ψ. An exact analogue of this theorem was proved in [6] (Theorem 3.9) where g and h were assumed to be bounded. Nevertheless, we present the proof here for the convenience of the reader.…”