In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on ${\mathbb{C}}^n$. 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.

中文翻译：

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海森堡群上的傅立叶变换的海森堡唯一性对

在本文中，我们证明（单位球面，非调和锥）是Heisenberg唯一对上的辛傅里叶变换 ${\mathbb{C}}^{ñ}$。我们推论，球体以及非谐波圆锥体都是确定单位球体上有限度量的频谱投影的确定集。此外，我们证明，如果在第二步幂等李群上的有限支持函数的傅立叶变换具有任意有限秩，则该函数必须为零。后者的结果与Weyl变换的the灭对相关。