An analogue of Ingham's theorem on the Heisenberg group

TL;DR

This paper establishes a Heisenberg group analogue of Ingham's uncertainty principle by constructing specific functions with controlled Fourier decay and extending Chernoff's theorem to special Hermite operators.

Contribution

It introduces a novel uncertainty principle for the Heisenberg group and develops new techniques for operator-valued Fourier transforms and special Hermite operators.

Findings

Constructed compactly supported functions with prescribed Fourier decay.

Proved an Ingham-type uncertainty principle on the Heisenberg group.

Extended Chernoff's theorem to special Hermite operators.

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.