  Fourier-positivity, i.e. the mathematical property that a function has a
positive Fourier transform, can be used as a constraint on the parametrization
of QCD dipole-target cross-sections or Wilson line correlators in transverse
position (r) space. They are Bessel transforms of positive transverse momentum
dependent gluon distributions. Using mathematical Fourier-positivity
constraints on the limit r -> 0 behavior of the dipole amplitudes, we identify
the common origin of the violation of Fourier-positivity for various, however
phenomenologically convenient, dipole models. It is due to the behavior
r^{2+epsilon}, epsilon>0, softer, even slightly, than color transparency.
Fourier-positivity seems thus to conflict with the present dipole formalism
when it includes a QCD running coupling constant alpha(r).
