The Rate of Convergence of Hermite Function Series

Let a > 0 be the least upper bound of y for which f(z) 0(e-°My) for some positive constant q as |z| -+ o» on the real axis. It is then proved that at least an infinite subsequence of the coefficients \an} in oo f(z) = e-z2/2 £ anHn(z), n=0 where the Hn 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 p for an entire function or the location of the nearest singularity for a singular function as for a power series but also upon a, 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(-'/îZ2) the poorer the convergence of the Hermite series will be.