Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may equivalently consider $$ f = a_0 + \sum_{n \geq 1} a_n H_n, \quad a \in \ell^2 $$
My questions are about the properties of the expansion above.
Say $a \in \ell^2$, and define $f_{N} = a_0 + \sum_{n \leq N} a_n H_n$
(1) What conditions are required for $a \in \ell^2$, to upgrade the convergence $f_N \stackrel{L^2(\gamma)}{\longrightarrow} f$ to pointwise almost everywhere as $N \to \infty$?
(2) Whenever pointwise a.e. convergence holds, when is the resulting $f$ uniformly bounded? I.e., based on the coefficients $a$, when is $f \in L^\infty(\gamma)$?
Regarding (2), a partial observation is that if $a \in c_{00}$, the space of sequences which are eventually zero, then this can only happen if $a_n = 0$ for $n \geq 1$ (simply because in this case $f$ is otherwise a polynomial with nontrivial degree and must therefore diverge on the left or right). This doesn't seem to indicate much about general $a \in \ell^2$, however, since there are uniformly bounded functions with infinite Hermite expansions.
