An analogue of Ingham’s theorem on the Heisenberg group

Abstract

We prove an exact analogue of Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose operator valued Fourier transforms have suitable Ingham type decay and proving an analogue of Chernoff's theorem for the family of special Hermite operators.

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

“…Hence by the remark after Theorem 3.1 we know that polynomials that are even in the second variable are dense in L 1 2,e (R 2 , dμ f ). Consider the function ϕ defined on α by ϕ(λ, (2k…”

“…Note added in the proof. The proof of Theorem 1.4 presented in [1] is not com-plete. As a consequence, the converse part of Theorem 1.6 remains unproved.…”

“…As a consequence, the converse part of Theorem 1.6 remains unproved. For the correct version we refer the reader to [1].…”