For each $n \in \mathbb{N}$, the Hermite function $\psi\_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi\_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}
\end{equation}
according to the Wikipedia article : <https://en.wikipedia.org/wiki/Hermite_polynomials#Differential-operator_representation>


Moreover, it is written there that each Hermite function satisfies the $L^\infty$ bound
\begin{equation}
\lvert \psi\_n(x) \rvert \leq \pi^{-1/4} \text{ for all } x \in \mathbb{R}, n \in \mathbb{N}
\end{equation}
which is known as the Cramer's inequality.


Now, I am trying to find a bound of
\begin{equation}
\sup\_{x \in \mathbb{R}} \lvert x^m \psi\_n(x) \rvert
\end{equation}
for any given $n,m \in \mathbb{N}$.


More specifically, I am curious about the asymptotic behavior of this supremum as $n \to \infty$ while $m$ is fixed. Would it remain bounded or diverge?


I searched for references myself but cannot get an idea about resolving this issue. Could anyone please help me?



$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
$\newcommand\ep\varepsilon$According to the 6th display in [this section](https://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion), for any fixed $\ep>0$ we have
$$\psi\_n(x)=(2/\pi)^{1/4}(\pi n)^{-1/4}(\sin t)^{-1/2}s\_n(x)$$
if $n\to\infty$, $x=\sqrt{2n+1}\,\cos t$, and $t\in[\ep,\pi-\ep]$, and
$$s\_n(x):=\sin\Big(\frac{3\pi}4+\frac{2n+1}4\,(\sin2t-2t)\Big)+O(1/n).$$


Fix now any $t\in(0,\pi/2)$ such that $\sin2t-2t$ is irrational, and let $x\_n:=\sqrt{2n+1}\,\cos t$. Then $\limsup\_n s\_n(x\_n)=1$ and hence
$$\limsup\_n x\_n^m\, s\_n(x\_n)=\infty$$
for each integer $m\ge1$.


So, for each integer $m\ge1$,
\begin{equation}
\sup\_{x\in\mathbb R}|x^m\,\psi\_n(x)|\text{ is unbounded in $n$}. 
\end{equation}




---


Using [Szegő, formula (8.22.14)](https://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf) (say with $t=0$), we see that, moreover, for each real $m>1/12$,
\begin{equation}
\lim\_n\sup\_{x\in\mathbb R}|x|^m\,|\psi\_n(x)|=\infty. 
\end{equation}


