A FREQUENCY SPACE FOR THE HEISENBERG GROUP

Hajer Bahouri (1) , Jean-Yves Chemin (2) , Raphael Danchin (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LJLL - Laboratoire Jacques-Louis Lions

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by taking advantage of basic properties of Hermite functions) the Fourier transform f_H of f to be a uniformly continuous mapping on the set N^d × N^d ×R \ {0} endowed with a suitable distance. This enables us to extend f_H to the completion of that space, and to get an explicit asymptotic description of the Fourier transform when the 'vertical' frequency tends to 0. We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the Euclidean case that are based on Fourier analysis. As an example, we here establish an explicit extension of the Fourier transform for smooth functions on H^d that are independent of the vertical variable.

Mots clés: Fourier transform; Heisenberg group; frequency space; Hermite functions AMS Subject Classification (2000): 43A30; 43A80

Domaines: Analyse classique [math.CA]

Hajer Bahouri, Jean-Yves Chemin, Raphael Danchin. A FREQUENCY SPACE FOR THE HEISENBERG GROUP. Annales de l'Institut Fourier, 2018, 69 (1), pp.365-407. ⟨hal-01363426⟩