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P. Roquette
1967
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J.–P. Serre
1971
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J. V. Armitage
1972
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A.M. Odlyzko
1977
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2008

Principe d'Heisenberg et fonctions positives

Bourgain, Jean, Clozel, Laurent, Kahane, Jean-Pierre

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Starting from a problem in number theory, the article investigates the properties of the couples of Fourier transforms on the real line, f and g, real and even, f >= 0 out of an interval (-a, a) and f(0) < 0, g >= 0 out of an interval (-b, b) and g(0) < 0 .

How small the product ab can be ? There is a strictly positive lower bound, the exact value is not known.
The same problem is considered in several dimensions (where it is related to number theory, as the article points out).