Here's a simple and far from optimal condition guaranteeing unifolrm convergence.

Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$  $$ \int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n. $$ In this case the associated Hermite series is  $$ \sum_{n\geq 0} \frac{c_n}{n!} H_n(x), $$ where $$ c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx. $$ and this converges to $f$ uniformly on compacts. This follows from known asymptotic estimates for Hermite polynomials.

For more precise results you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.

Here's a simple and far-from-optimal condition guaranteeing uniform convergence.

Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$ 

$$ \int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n. $$

In this case the associated Hermite series is 

$$ \sum_{n\geq 0} \frac{c_n}{n!} H_n(x), $$

where

$$ c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx. $$

and this converges to $f$ uniformly on compacts. This follows from known asymptotic estimates for Hermite polynomials.

For more precise results, you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.

Here's a simple and far from optimal condition guaranteeing unifolrm convergence.

Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$ $$ \int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n. $$ In this case the associated Hermite series is $$ \sum_{n\geq 0} \frac{c_n}{n!} H_n(x), $$ where $$ c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx. $$ and this converges to $f$ uniformly on compacts. Form more precise results you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.

Here's a simple and far from optimal condition guaranteeing unifolrm convergence.

Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$ $$ \int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n. $$ In this case the associated Hermite series is $$ \sum_{n\geq 0} \frac{c_n}{n!} H_n(x), $$ where $$ c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx. $$ and this converges to $f$ uniformly on compacts. This follows from known asymptotic estimates for Hermite polynomials.

For more precise results you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.