  Characterizing in a constructive way the set of real functions whose Fourier
transforms are positive appears to be yet an open problem. Some sufficient
conditions are known but they are far from being exhaustive. We propose two
constructive sets of necessary conditions for positivity of the Fourier
transforms and test their ability of constraining the positivity domain. One
uses analytic continuation and Jensen inequalities and the other deals with
Toeplitz determinants and the Bochner theorem. Applications are discussed,
including the extension to the two-dimensional Fourier-Bessel transform and the
problem of positive reciprocity, i.e. positive functions with positive
transforms.
