  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).
