  Numerical approximation of the Boltzmann equation is a challenging problem
due to its high-dimensional, nonlocal, and nonlinear collision integral. Over
the past decade, the Fourier-Galerkin spectral method has become a popular
deterministic method for solving the Boltzmann equation, manifested by its high
accuracy and potential of being further accelerated by the fast Fourier
transform. Albeit its practical success, the stability of the method is only
recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363,
no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the
collision operator. In this work, we provide a new proof based on a careful
$L^2$ estimate of the negative part of the solution. We also discuss the
applicability of the result to various initial data, including both continuous
and discontinuous functions.
