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A frequency space for the Heisenberg group. (Un espace de fréquences pour le groupe de Heisenberg.) (English. French summary) Zbl 1416.43005
Summary: We revisit the Fourier analysis on the Heisenberg group $$\mathbb{H}^d$$. Starting from the so-called Schrödinger representation and taking advantage of the projection with respect to the Hermite functions, we look at the Fourier transform of an integrable function $$f$$, as a function $$\widehat{f}_\mathbb{H}$$ on the set $$\widetilde{\mathbb{H}}^d\mathop {=}\limits ^{\text{def}}\mathbb{N}^d\times \mathbb{N}^d\times \mathbb{R}\setminus \lbrace 0\rbrace$$. After observing that $$\widehat{f}_{\mathbb{H}}$$ is uniformly continuous on $$\widetilde{\mathbb{H}}^d$$ equipped with an appropriate distance $$\widehat{d},$$ we extend the definition of $$\widehat{f}_\mathbb{H}$$ to the completion $$\widehat{\mathbb{H}}^d$$ of $$\widetilde{\mathbb{H}}^d$$. This new point of view provides a simple and explicit description of the Fourier transform of integrable functions, when the “vertical” frequency parameter tends to $$0$$. As an application, we prepare the ground for computing the Fourier transform of functions on $$\mathbb{H}^d$$ that are independent of the vertical variable.

##### MSC:
 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A80 Analysis on other specific Lie groups
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##### References:
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