Principe d’Heisenberg et fonctions positives
Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1215-1232.

On décrit un problème naturel concernant la transformation de Fourier. Soient f, f^ deux fonctions associées par celle-ci, positives pour xa et nulles en zéro. Quelle est la borne inférieure pour a ? En dimension supérieure, même question, l’intervalle étant remplacé par la boule de rayon a. On montre l’existence d’une borne inférieure strictement positive, qui est estimée en fonction de la dimension. La dernière section montre que cette question est naturellement liée à la théorie des fonctions zêta.

We consider a natural problem concerning Fourier transforms. In one variable, one seeks functions f and f^, both positive for xa and vanishing at 0. What is the lowest bound for a ? In higher dimension, the same problem can be posed by replacing the interval by a ball of radius a. We show that there is indeed a strictly positive lower bound, which is estimated as a function of the dimension. In the last section the question, and its solution, are shown to be naturally related to the theory of zêta-functions.

DOI : 10.5802/aif.2552
Classification : 42A38, 42B10, 11R42
Mot clés : transformation de Fourier, fonctions zêta
Keywords: Fourier transforms, zêta-function
Bourgain, Jean 1 ; Clozel, Laurent 2 ; Kahane, Jean-Pierre 2

1 School of Mathematics Institute for Advanced Study Princeton, N.J. 08540 (USA)
2 Laboratoire de Mathématique Université Paris–Sud, Bât. 425 91405 Orsay Cedex (France)
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Bourgain, Jean; Clozel, Laurent; Kahane, Jean-Pierre. Principe d’Heisenberg et fonctions positives. Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1215-1232. doi : 10.5802/aif.2552. http://www.numdam.org/articles/10.5802/aif.2552/

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