Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The rate of convergence of Hermite function series

Author: John P. Boyd
Journal: Math. Comp. 35 (1980), 1309-1316
MSC: Primary 42C10
MathSciNet review: 583508
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\alpha > 0$ be the least upper bound of $\gamma$ for which \[ f(z) \sim O({e^{ - q|z|\gamma }})\] for some positive constant q as $|z| \to \infty$ 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 $\exp (-1/2 z^2)$ the poorer the convergence of the Hermite series will be.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 42C10

Retrieve articles in all journals with MSC: 42C10

Additional Information

Keywords: Hermite function series
Article copyright: © Copyright 1980 American Mathematical Society