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In this article, we prove that if the group Fourier transform of certainintegrable functions on the Heisenberg motion group (or step two nilpotent Liegroups) is of finite rank, then the function is identically zero. These resultscan be thought as an analogue to the Benedicks theorem that dealt with theuniqueness of the Fourier transform of integrable functions on the Euclideanspaces.

Uniqueness of the group Fourier transform on certain nilpotent Lie groups