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Mathematics of Computation
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
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Abstract: Let $ \alpha > 0$ be the least upper bound of $ \gamma $ for which

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

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

$\displaystyle 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0583508-3
PII: S 0025-5718(1980)0583508-3
Keywords: Hermite function series
Article copyright: © Copyright 1980 American Mathematical Society