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.

  • 1
    $\begingroup$ In the book: Deitmar and Echterhoff: Principles of Harmonic Analysis, this concept is called "Dirac net", or in the case of Lie groups, "Dirac sequence". In the context of Banach algebras, it often goes as "approximate identity". $\endgroup$
    – user1688
    Mar 17 '14 at 12:35
  • $\begingroup$ Thanks Anton. I have since found something like this in Loomis "An Introduction to Abstract Harmonic Analysis" (in the context of $L_p$ functions and convolution algebras on groups). I was hoping that the approximation of continuous functions by smooth ones through convolution in Lie groups was already explicitly written somewhere classic and not have to write the simple proof myself. In Spanish one says "Lazy people work twice" ... $\endgroup$ Mar 17 '14 at 12:44
  • $\begingroup$ The question is clearly local, so couldn't you simply refer to this fact in the Euclidean case? $\endgroup$ Mar 18 '14 at 19:01
  • $\begingroup$ It's not a question of using any sort of convolution to smooth the function. I need the convolutions to be of this special form because the approximating functions have to satisfy special properties that are preserved by the group action. $\endgroup$ Mar 18 '14 at 19:08

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.


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