Say we have the following set of functions
$$\mathcal S = \{f_1,f_2,\cdots\}$$
so that for any $n$ there exist uniquely a set of $ \{ c_k \}$:
$$(f_n * g)(t) = \sum_{k\in \mathbb Z^+} c_k f_k(t)$$
(or maybe normalized, but that is not of importance here)
, where $g(t)$ is a gaussian function: $$g(t) = \exp\left(\frac{-t^2}{\sigma^2}\right)$$
How can I find such sets $\mathcal S$ ?
Can I for example prove or disprove whether the set of monomials $f_k(t) = t^k$ is such a space?