A frequency space for the Heisenberg group
[Un espace de fréquences pour le groupe de Heisenberg]
Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 365-407.

On propose une nouvelle approche de la théorie de Fourier sur le groupe de Heisenberg. En utilisant la représentation de Schrödinger et la projection sur les fonctions de Hermite, la transformée de Fourier d’une fonction intégrable est définie comme une fonction sur l’ensemble  ˜ d = def d × d ×{0}. Cette fonction étant uniformément continue sur  ˜ d muni d’une distance adéquate, on peut l’étendre par densité sur le complété  ^ d de  ˜ d . Ce nouveau point de vue fournit une description simple de la limite de la transformée de Fourier des fonctions intégrables lorsque la « fréquence verticale » tend vers 0. On dispose ainsi d’un cadre adéquat pour calculer par exemple la transformée de Fourier d’une fonction indépendante de la variable verticale.

We revisit the Fourier analysis on the Heisenberg group 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 f ^ on the set  ˜ d = def d × d ×{0}. After observing that f ^ is uniformly continuous on  ˜ d equipped with an appropriate distance d ^, we extend the definition of f ^ to the completion ^ d of ˜ 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  d that are independent of the vertical variable.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3246
Classification : 43A30,  43A80
Mots clés : Transformée de Fourier, groupe de Heisenberg, espace des fréquences, fonctions de Hermite.
@article{AIF_2019__69_1_365_0,
     author = {Bahouri, Hajer and Chemin, Jean-Yves and Danchin, Rapha\"el},
     title = {A frequency space for the Heisenberg group},
     journal = {Annales de l'Institut Fourier},
     pages = {365--407},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {1},
     year = {2019},
     doi = {10.5802/aif.3246},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3246/}
}
Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël. A frequency space for the Heisenberg group. Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 365-407. doi : 10.5802/aif.3246. http://archive.numdam.org/articles/10.5802/aif.3246/

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