Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

Abstract

In this article, we prove that (non-harmonic cone, unit sphere) is a Heisenberg uniqueness pair for the symplectic Fourier transform on C n . And we derive that a sphere whose radius is not contained in the zero sets of the Laguerre polynomials is a determining set for the spectral projections corresponding to the finite measure supported on the unit sphere. Further, we prove that if the Fourier transform of a certain finitely supported function on step two nilpotent Lie groups is of arbitrary finite rank, then the function must be zero.

Benedicks-Amrein-Berthier theorem for the Heisenberg motion group and quaternion Heisenberg group

“…In [5] authors consider B as a rectangle in R 2n while proving an analogous result on step two nilpotent Lie groups and a version of this result, with a strong assumption on rank, derived therein for the Heisenberg motion group. Later in the article [9], this result is extended to arbitrary set B of finite measure for general step two nilpotent Lie groups. We prove the result on the Heisenberg motion group and quaternion Heisenberg group when B is an arbitrary set of finite measure using the Hilbert space theory.…”