  This paper presents a dissipativeness analysis of a quadrature method of
moments (called HyQMOM) for the one-dimensional BGK equation. The method has
exhibited its good performance in numerous applications. However, its
mathematical foundation has not been clarified. Here we present an analytical
proof of the strict hyperbolicity of the HyQMOM-induced moment closure systems
by introducing a polynomial-based closure technique. As a byproduct, a class of
numerical schemes for the HyQMOM system is shown to be realizability preserving
under CFL-type conditions. We also show that the system preserves the
dissipative properties of the kinetic equation by verifying a certain
structural stability condition. The proof uses a newly introduced affine
invariance and the homogeneity of the HyQMOM and heavily relies on the theory
of orthogonal polynomials associated with realizable moments, in particular,
the moments of the standard normal distribution.
