Phase-space consistency of stellar dynamical models determined by
  separable augmented densities
  Assuming the separable augmented density, it is always possible to construct
a distribution function of a spherical population with any given density and
anisotropy. We consider under what conditions the distribution constructed as
such is in fact non-negative everywhere in the accessible phase-space. We first
generalize known necessary conditions on the augmented density using fractional
calculus. The condition on the radius part R(r^2) (whose logarithmic derivative
is the anisotropy parameter) is equivalent to the complete monotonicity of
R(1/w)/w. The condition on the potential part on the other hand is given by its
derivative up to any order not greater than (3/2-beta) being non-negative where
beta is the central anisotropy parameter. We also derive a specialized
inversion formula for the distribution from the separable augmented density,
which leads to sufficient conditions on separable augmented densities for the
non-negativity of the distribution. The last generalizes the similar condition
derived earlier for the generalized Cuddeford system to arbitrary separable
systems.
