  For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$
a compact subgroup, the "Schwartz correspondence" states that the spherical
transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash
G/K)$ isomorphically onto the space ${\mathcal S}(\Sigma_{\mathcal D})$, where
$\Sigma_{\mathcal D}$ is an embedded copy of the Gelfand spectrum in ${\mathbb
R}^\ell$, canonically associated to a generating system ${\mathcal D}$ of
$G$-invariant differential operators on $G/K$, and ${\mathcal
S}(\Sigma_{\mathcal D})$ consists of restrictions to $\Sigma_{\mathcal D}$ of
Schwartz functions on ${\mathbb R}^\ell$.
  Schwartz correspondence is known to hold for a large variety of Gelfand pairs
of polynomial growth. In this paper we prove that it holds for the strong
Gelfand pair $(M_n,SO_n)$ with $n=3,4$. The rather trivial case $n=2$ is
included in previous work by the same authors.
