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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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