Particular solutions of the inverse problem for 1D Vlasov-Maxwell equilibria using Hermite polynomials

We present the solution to an inverse problem arising in the context of finding a distribution function for a specific collisionless plasma equilibrium. The inverse problem involves the solution of two integral equations, each having the form of a Weierstrass transform. We prove that inverting the Weierstrass transform using Hermite polynomials leads to convergent infinite series. We also comment on the non-negativity of the distribution function, with more detail on this in Allanson $\textit{et al., Journal of Plasma Physics}$, vol. 82 (03), 2016. Whilst applied to a specific magnetic field, the inversion techniques used in this paper (as well as the derived convergence criteria and discussion of non-negativity) are of a general nature, and are applicable to other smooth pressure functions.