Let $V$ be a finite dimensional euclidean space and let $G\subset O(V)$ be a finite (or compact) group, let $\mathbb{R}[V]$ be the algebra of polynomial functions on $V$. If $E\subset \mathcal{C}^\infty(V)$ is a sub algebra, we denote by $E^G$ the sub-algebra of $G$-invariant function of $E$.

A [theorem of Schwarz](https://doi.org/10.1016/0040-9383(75)90036-1 "Schwarz - Smooth functions invariant under the action of a compact Lie group") says:

**Theorem :** Let $(P_1,\dotsc,P_k)$ be a system of generators of $\mathbb{R}[V]^G$ as an algebra over $\mathbb{R}$, then $\mathcal{C}^\infty(V)^G$ is "smoothly generated" by $(P_1,\dotsc,P_k)$ in the sense that the map:

$$\begin{align}\mathcal{C}^\infty(\mathbb{R}^k)&\to \mathcal{C}^\infty(V)^G\\\\ g&\mapsto g(P_1,\dots,P_k)
\end{align}$$
is surjective (and continuous in the right topologies).

There is also [another proof by Bierstone](https://doi.org/10.4310/jdg/1214433159 "Bierstone - Local properties of smooth maps equivariant with respect to finite group actions") in case $G$ is finite (although the proof only works at the level of germs of smooth functions if I understand correctly).

My question relates to the extension of this kind of results to subalgebras of smooth functions.

> **Question**
>
> Let $E\subset \mathcal{C}^\infty(V)$ be such that $(E\cap \mathbb{R}[V])^G$ is generated as an algebra over $\mathbb{R}$ by $(Q_1,\dots,Q_k)$, under what conditions on $E$ and $G$ is $E^G$
>"smoothly generated" by $(Q_1,\dotsc,Q_k)$ ?

I'm currently trying to read through the proofs of Schwarz and Bierstone but have yet to understood if and how they can be adapted to this setting. The case of a finite group $G$ is already very interesting for what I have in mind. If this helps, one can also assume that $E$ is actually an ideal.

An obvious restriction should be that $E$ and $E^G$ contain enough polynomials, and probably density of $\mathbb{R}[V]\cap E$ in $E$ at least for the $\mathcal{C}_\text{loc}^0$ topology should be required.


[1]: https://core.ac.uk/download/pdf/81994047.pdf
[2]: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-10/issue-4/Local-properties-of-smooth-maps-equivariant-with-respect-to-finite/10.4310/jdg/1214433159.full