I'm looking for a nice (and preferably classic or book) reference for the following *type* of result:

Consider a transitive action of a compact Lie group $G$ on a compact manifold $M$ and a continuous function $f : M \rightarrow \mathbb{R}$. If $\phi_n : G \rightarrow \mathbb{R}$ is a sequence of smooth functions such that

**1.** The support of $\phi_{n + 1}$ is contained in the support of $\phi_n$ and the intersection of the supports of all these functions is the identity $e \in G$.

**2.** The integral of $\phi_n$ over $G$ with respect to the Haar measure is equal to $1$ for every value of $n$.

**3.** (optional) $\phi_n \geq 0$ for all values of $n$

Then the functions
$$
f_n (x) := \int_{g \in G} f(g^{-1} \cdot x) \phi_n(g) \, dg
$$
form a sequence of *smooth* functions converging uniformly to $f$.

This reduces to the particular case where $G = M$, the action is group multiplication, and this is the "standard" construction with convolution, but I can't think of good reference to cite.