I would like to understand $L^2\left(\mathbb{R}, \mu\right)$ approximation by polynomials of $\tanh$ (and more generally smooth functions), where $\mu$ is standard gaussian distribution. This leads to considering the Hermite expansion of $\tanh$. We want to look at the coefficients $a_k$ in the Hermite expansion of $\tanh(x) = \sum_{k \geq 0} a_k He_k(x)$, where \begin{equation*} a_{k}=\int_{-\infty}^{\infty} He_{k}(x) e^{-x^{2} / 2} \tanh (x) d \mu \end{equation*} and $He_k$ denotes the $k$th normalized probabilists' Hermite polynomial (https://en.wikipedia.org/wiki/Hermite_polynomials#Definition).
Using Theorem 1 in a paper of E. Hille (https://www.ams.org/journals/tran/1940-047-01/S0002-9947-1940-0000871-3/S0002-9947-1940-0000871-3.pdf), we can show that the magnitude of $k$-th coefficient is bounded by $e^{-\Omega\left(\frac{\pi\sqrt{k}}{4}\right)}$.
Also, using Theorem 2 in a paper of J. P. Boyd (https://deepblue.lib.umich.edu/bitstream/handle/2027.42/24797/0000223.pdf?sequence=1&isAllowed=y), we can show that the magnitude of an infinite subsequence of the coefficients $\left\{a_k\right\}$ in the Hermite expansion of $\tanh$ will be at least $c e^{-dk^{0.5 + \epsilon}}$, for positive constants $c$ and $d$ and an arbitrarily small constant $\epsilon$>0.
What can we say about the lower bound for all the non-zero coefficients in the Hermite expansion of $\tanh$? Specifically, how dense is the subsequence of the "large" coefficients as in Boyd's theorem above? Table III in another research paper of John P. Boyd (https://deepblue.lib.umich.edu/bitstream/handle/2027.42/24797/0000223.pdf?sequence=1&isAllowed=y) shows that the formula
\begin{equation*}
a_k \approx \frac{2}{(2k + 1)^{1/4}} e^{-p \sqrt{2n+1}}
\end{equation*}
approximates nearly all the non zero coefficients of $\mathbb{sech}$. However, the derivation of the formula is for the asymptotic case, i.e. $k \to \infty$. We would like non-asymptotic bounds.
