  Let $V$ be a real finite dimensional representation of a compact Lie group
$G$. It is well-known that the algebra $\mathbb R[V]^G$ of $G$-invariant
polynomials on $V$ is finitely generated, say by $\sigma_1,...,\sigma_p$.
Schwarz proved that each $G$-invariant $C^\infty$-function $f$ on $V$ has the
form $f=F(\sigma_1,...,\sigma_p)$ for a $C^\infty$-function $F$ on $\mathbb
R^p$. We investigate this representation within the framework of
Denjoy-Carleman classes. One can in general not expect that $f$ and $F$ lie in
the same Denjoy-Carleman class $C_M$ (with $M=(M_k)$). For finite groups $G$
and (more generally) for polar representations $V$ we show that for each
$G$-invariant $f$ of class $C_M$ there is an $F$ of class $C_N$ such that
$f=F(\sigma_1,...,\sigma_p)$, if $N$ is strongly regular and satisfies $N_k \ge
M_{km} \ep^{k+1}$, for all $k$, with $m$ an (explicitly known) integer
depending only on the representation and $\epsilon>0$ independent of $k$. In
particular, each $G$-invariant $(1+\delta)$-Gevrey function $f$ has the form
$f=F(\sigma_1,...,\sigma_p)$ for a $(1+\delta m)$-Gevrey function $F$.
Applications to equivariant functions and basic differential forms are given.
