**tl;dr:** Are there known convergence estimates for approximating a function with a radial basis family?

**Details:** Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\tfrac1{2\sigma^2}(x-\mu)^2) : \mu\in\mathbb{R}, \sigma\ge 0\}$ and let $\mathcal{G}_{m,\alpha}$ denote the set of all linear combinations of functions from $\mathcal{G}$ with at most $m$ terms and whose nonzero coefficients are bounded away from $\alpha$ (for completeness, $\alpha=0$ is allowed):
$$
\mathcal{G}_{m,\alpha}
= \Bigg\{\sum_{k=1}^m a_k g_k : g_k\in\mathcal{G},\, |a_k|\ge\alpha\cdot1(a_k\ne 0)\Bigg\}.
$$

Let also $\mathcal{F}$ be a (sufficiently regular) family of functions. Let $f\in\mathcal{F}$ and $g^*_{m,\alpha}$ be such that $$ d(f,g^*_{m,\alpha}) = \inf_{g\in\mathcal{G}_{m,\alpha}} d(f,g), $$

where $d$ is some metric on functions. (I am including the case $\alpha=0$ for this question. The reason for considering $\alpha > 0$ is that I am interested in avoiding pathologies with weights that tend to zero in the above projection as $m$ increases.)

I am interested in bounds of the form $d(f,g^*_{m,\alpha})\le C/m^\beta$ for some constant $C$ (which may or may not depend on $f$ and $\alpha$--I am mostly focused on the dependence on $m$). My questions are as follows:

- In the first place, are there known bounds of this form for nontrivial choices of $(d,\mathcal{F},C)$? I am being deliberately vague here in order to understand what types of results along these lines are available.
- Ideally, I am interested in the case where $\mathcal{F}$ is all smooth densities on $\mathbb{R}$, $\mathcal{G}_{m,\alpha}$ is restricted to
*convex combinations*of Gaussians, $\alpha>0$, and $d$ is a standard probability metric such as Hellinger or total variation.

**Some additional background:** These types of results are quite standard in the approximation theory literature when $\mathcal{G}$ is a set of algebraic or trigonometric polynomials (e.g. Bernstein approximation). Of course, it is well-known that $d(f,g^*_m)\to0$ when $\mathcal{G}$ is the set of Gaussian densities, however, I am not aware of any quantitative convergence estimates for this family. The one paper I am aware of is Li and Barron (2000) who consider the case $\alpha=0$.