  We consider the problem of approximating a general Gaussian location mixture
by finite mixtures. The minimum order of finite mixtures that achieve a
prescribed accuracy (measured by various $f$-divergences) is determined within
constant factors for the family of mixing distributions with compactly support
or appropriate assumptions on the tail probability including subgaussian and
subexponential. While the upper bound is achieved using the technique of local
moment matching, the lower bound is established by relating the best
approximation error to the low-rank approximation of certain trigonometric
moment matrices, followed by a refined spectral analysis of their minimum
eigenvalue. In the case of Gaussian mixing distributions, this result corrects
a previous lower bound in [Allerton Conference 48 (2010) 620-628].
