TITLE: Approximations of the identity on Lie groups and homogenous spaces
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: I found the reference to what I was looking for in Section 2 of the paper Convolutions, transforms and convex bodies by Eric Grinberg and Gaoyong Zhang. They use the technique to prove that bodies of constant width or brightness can be approximated by smooth or even real-analytic bodies of constant width or brightness, as well as many other neat results.