Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

2018/10/11 by Somnath Ghosh, Ghosh, Somnath, R. K. Srivastava +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1810.06390

openalex publication_date 2018/10/11 · openalex created_date 2018/10/26 · openalex updated_date 2026/07/28

Abstract

In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on \mathbb Cⁿ. We derive that spheres as well as non-harmonic cones are determining sets for the spectral projections of the finite measure supported on the unit sphere. Further, we prove that if the Fourier transform of a finitely supported function on step two nilpotent Lie group is of arbitrary finite rank, then the function must be zero. The latter result correlates to the annihilating pair for the Weyl transform.

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

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