The rate of convergence of Hermite function series

Abstract

Let α > 0 \alpha > 0 be the least upper bound of γ \gamma for which \[ f ( z ) ∼ O ( e − q | z | γ ) f(z) \sim O({e^{ - q|z|\gamma }}) \] for some positive constant q as | z | → ∞ |z| \to \infty on the real axis. It is then proved that at least an infinite subsequence of the coefficients { a n } \{ {a_n}\} in \[ f ( z ) = e − z 2 / 2 ∑ n = 0 ∞ a n H n ( z ) , f(z) = {e^{ - {z^2}/2}}\sum \limits _{n = 0}^\infty {{a_n}{H_n}(z),} \] where the H n {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 ρ \rho for an entire function or the location of the nearest singularity for a singular function as for a power series but also upon α \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 ) f(z) and exp ⁡ ( − 1 / 2 z 2 ) \exp (-1/2 z^2) the poorer the convergence of the Hermite series will be.