  The weights $W_\alpha(x)=\exp{(-|x|^{\alpha})}$ $(\alpha>1)$ form a subclass
of Freud weights on the real line. Primarily from a functional analytic angle,
we investigate the subspace of $L_2(\mathbb R, W_\alpha^2(x)\,dx)$ consisting
of those elements that can be rapidly approximated by polynomials. This
subspace has a natural Fr\'echet topology, in which it is isomorphic to the
space of rapidly decreasing sequences. We show that it consists of smooth
functions and obtain concrete results on its topology. For $\alpha=2$ there is
a complete and elementary description of this topological vector space in terms
of the Schwartz functions.
