ca.analysis and odes - Space of rapidly decreasing functions

It's (still) not completely clear what space you are working in since you haven't clarified the confusions from Yemon Choi and fedja's comments. But it may help to note an analogous problem that is well-defined and where there is a clear strategy. That strategy may adapt to your situation (whatever that is).

This analogous situation is of periodic functions using the usual basis of periodic exponentials, $e^{2pi i n x}$. The key here is:

$g$ is $C^infty$ if and only if the Fourier coefficients $c_n(g) = int_0^1 g(x) e^{-2pi i n x} d x$ are rapidly decreasing.

Then for $g$ in $L^2$, $c_n(g)$ is square summable and so $c_n(g) e^{-n t}$ for $t > 0$ is rapidly decreasing (that is, $lim_{n to infty} n^k c_n(g) e^{-n t} = 0$ for each $k$) since exponentials beat polynomials.

So I would try to find out a characterisation of the Schwartz space of rapidly decreasing functions in terms of the coefficients of their expansions with respect to the Hermite polynomials (suitably weighted). I would expect such a characterisation to be well-known, but I don't know it off the top of my head.