A frequency space for the Heisenberg group
Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 365-407.

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.

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.

Published online:
DOI: 10.5802/aif.3246
Classification: 43A30, 43A80
Keywords: Fourier transform, Heisenberg group, frequency space, Hermite functions.
Mot clés : Transformée de Fourier, groupe de Heisenberg, espace des fréquences, fonctions de Hermite.
Bahouri, Hajer 1; Chemin, Jean-Yves 2; Danchin, Raphaël 1

1 LAMA, UMR 8050 Univ. Paris-Est Créteil, 94010 Créteil Cedex (France)
2 Laboratoire J.-L. Lions, UMR 7598 Sorbonne Université, 75230 Paris Cedex 05 (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël. A frequency space for the Heisenberg group. Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 365-407. doi : 10.5802/aif.3246. https://aif.centre-mersenne.org/articles/10.5802/aif.3246/

[1] Astengo, Francesca; Di Blasio, Bianca; Ricci, Fulvio Fourier transform of Schwartz functions on the Heisenberg group, Stud. Math., Volume 214 (2013) no. 3, pp. 201-222 | DOI | MR | Zbl

[2] Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël Fourier transform of tempered distributions on the Heisenberg group (2017) (https://arxiv.org/abs/1705.02195, to appear in Ann. Henri Lebesgue) | Zbl

[3] Bahouri, Hajer; Fermanian-Kammerer, Clotilde; Gallagher, Isabelle Phase-space analysis and pseudodifferential calculus on the Heisenberg group, Astérisque, 342, Société Mathématique de France, 2012, vi+127 pages | MR | Zbl

[4] Beals, Richard; Greiner, Peter Calculus on Heisenberg manifolds, Annals of Mathematics Studies, 119, Princeton University Press, 1988, x+194 pages | DOI | MR | Zbl

[5] Ciatti, Paolo; Ricci, Fulvio; Sundari, Maddala Uncertainty inequalities on stratified nilpotent groups, Bull. Kerala Math. Assoc., Volume 3 (2006) no. Special Issue, pp. 53-72 | MR | Zbl

[6] Corwin, Lawrence J.; Greenleaf, Frederick P. Representations of nilpotent Lie groups and their applications. Part I Basic theory and examples, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, 1990, viii+269 pages | MR | Zbl

[7] Faraut, Jacques; Harzallah, Khélifa Deux cours d’analyse harmonique, Progress in Mathematics, 69, Birkhäuser, 1987, viii+293 pages (Papers from the Tunis summer school held in Tunis, August 27–September 15, 1984) | MR | Zbl

[8] Fischer, Véronique; Ruzhansky, Michael A pseudo-differential calculus on graded nilpotent Lie groups, Fourier analysis. Pseudo-differential operators, time-frequency analysis and partial differential equation (Trends in Mathematics), Birkhäuser, 2014, pp. 107-132 | MR | Zbl

[9] Folland, Gerald B. Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, 1989, x+277 pages | DOI | MR | Zbl

[10] Geller, Daryl Fourier analysis on the Heisenberg group. I. Schwartz space, J. Funct. Anal., Volume 36 (1980) no. 2, pp. 205-254 | DOI | MR | Zbl

[11] Huet, Denise Décomposition spectrale et opérateurs, Le Mathématicien, 16, Presses Universitaires de France, 1976, 148 pages | MR | Zbl

[12] Hulanicki, Andrzej A functional calculus for Rockland operators on nilpotent Lie groups, Stud. Math., Volume 78 (1984) no. 3, pp. 253-266 | DOI | MR | Zbl

[13] Lakshmi Lavanya, R.; Thangavelu, Sundaram Revisiting the Fourier transform on the Heisenberg group, Publ. Mat., Barc., Volume 58 (2014) no. 1, pp. 47-63 http://projecteuclid.org/euclid.pm/1387570390 | DOI | MR | Zbl

[14] Olver, Frank William John Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, 1974, xvi+572 pages | MR | Zbl

[15] Ricci, Fulvio; Stein, Elias M. Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal., Volume 73 (1987) no. 1, pp. 179-194 | DOI | MR | Zbl

[16] Rudin, Walter Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons, 1990, x+285 pages (Reprint of the 1962 original) | DOI | MR | Zbl

[17] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993, xiv+695 pages (With the assistance of Timothy S. Murphy) | MR | Zbl

[18] Taylor, Michael E. Noncommutative harmonic analysis, Mathematical Surveys and Monographs, 22, American Mathematical Society, 1986, xvi+328 pages | DOI | MR | Zbl

[19] Thangavelu, Sundaram Some uncertainty inequalities, Proc. Indian Acad. Sci., Math. Sci., Volume 100 (1990) no. 2, pp. 137-145 | DOI | MR | Zbl

[20] Thangavelu, Sundaram Harmonic analysis on the Heisenberg group, Progress in Mathematics, 159, Birkhäuser, 1998, xiv+192 pages | DOI | MR | Zbl

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