Approximation theory in the context of probability density
function turns out to go beyond the classical idea of orthogonal
projection. Special tools have to be designed so as to respect the
nonnegativity of the approximate function. We develop here and
justify from the theoretical point of view an approximation
procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an
entropy minimization principle under moment constraints. We prove
in particular a global existence theorem for such an approximation
and derive as a by-product a necessary and sufficient condition
for the so-called problem of moment realizability.
Applications of the above result are given in kinetic theory:
first in the context of Levermore's approach and second to design
generalized BGK models for Maxwellian molecules.