Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function.

If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as

\begin{equation} \sum_{n=0}^\infty d_n H_n(x) \end{equation} where \begin{equation} d_n=\frac{1}{\sqrt{2\pi}n!}\int_{\mathbb{R}}f(x)H_n(x)e^{-x^2/2}dx \end{equation}

According to Convergence of orthogonal polynomial expansions, the Hermite expansion converges uniformly to $f$ if $\lVert Hf \rVert_{L^2} < \infty$.

However, if I just assume that $f$ is smooth and bounded, can I still say something about convergence of its Hermite expansion?

I tried to look for references myself, but I could not find anything relevant.