Consider the space $L^2(\mathbb{R})$. Let $(x_n)_n \subset \mathbb{R}$ be a sequence and $f \in L^2(\mathbb{R})$ a functions. It is well understood under which assumptions the span of the set $$ S = \{ f(\cdot - x_n) : n \in \mathbb{N} \} $$ is dense in $L^2(\mathbb{R})$. This goes into the theory of generators of $L^2(\mathbb{R})$. In particular, one can show that if $f$ is a Gaussian then we can find a sequence $(x_n)_n$ such that the span of $S$ is dense.

Can this be generalized to $L^2(\mathbb{R}^d)$ where $d > 1$? In this case, $(x_n)_n$ would be a sequence in $\mathbb{R}^d$. Is it again possible to choose $f$ to be a Gaussian? So far, I didn't find results on this question in the literature.