  We consider the theory and application of a solution method for the inverse
problem in collisionless equilibria, namely that of calculating a
Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using
Jeans' Theorem, the equilibrium distribution functions are expressed as
functions of the constants of motion, in the form of a Maxwellian multiplied by
an unknown function of the canonical momenta. In this case it is possible to
reduce the inverse problem to inverting Weierstrass transforms, which we
achieve by using expansions over Hermite polynomials. A sufficient condition on
the pressure tensor is found which guarantees the convergence and the
boundedness of the candidate solution, when satisfied. This condition is
obtained by elementary means, and it is clear how to put it into practice. We
also argue that for a given pressure tensor for which our method applies, there
always exists a positive distribution function solution for a sufficiently
magnetised plasma. Illustrative examples of the use of this method with both
force-free and non-force-free macroscopic equilibria are presented, including
the full verification of a recently derived distribution function for the
force-free Harris Sheet (Allanson $\textit{et al., Phys. Plasmas}$, vol. 22
(10), 2015, 102116). In the effort to model equilibria with lower values of the
plasma beta, solutions for the same macroscopic equilibrium in a new gauge are
calculated, with numerical results presented for $\beta_{pl}=0.05$.
