The rate of convergence of Hermite function series

Let $\alpha > 0$ be the least upper bound of $\gamma$ for which

$$f(z) \sim O({e^{ - q\vert z\vert\gamma }})$$

for some positive constant q as $\vert z\vert \to \infty$ on the real axis. It is then proved that at least an infinite subsequence of the coefficients $\{ {a_n}\}$ in

$$f(z) = {e^{ - {z^2}/2}}\sum\limits_{n = 0}^\infty {{a_n}{H_n}(z),}$$

where the ${H_n}$ are the normalized Hermite polynomials, must satisfy certain lower bounds. The theorems show two striking facts. First, the convergence rate of a Hermite series depends not only upon the order $\rho$ for an entire function or the location of the nearest singularity for a singular function as for a power series but also upon $\alpha$, thus making the convergence theory of Hermitian series more complicated (and interesting) than that for any ordinary Taylor expansion. Second, the poorer the match between the asymptotic behavior of $f(z)$ and $\exp ( - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {z^2})$ the poorer the convergence of the Hermite series will be.