The rate of convergence of Hermite function series
Author:
John P. Boyd
Journal:
Math. Comp. 35 (1980), 1309-1316
MSC:
Primary 42C10
DOI:
https://doi.org/10.1090/S0025-5718-1980-0583508-3
MathSciNet review:
583508
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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.
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M. ABRAMOWITZ & I. A. STEGUN, Eds., Handbook of Mathematical Functions, Dover, New York, 1965. MR 29 #4914.
J. P. BOYD, "Hermite polynomial methods for obtaining analytical and numerical solutions to eigenvalue problems in unbounded and spherical geometry," J. Comput. Phys. (Submitted.)
- Einar Hille, Contributions to the theory of Hermitian series, Duke Math. J. 5 (1939), 875–936. MR 870
- Einar Hille, Contributions to the theory of Hermitian series. II. The representation problem, Trans. Amer. Math. Soc. 47 (1940), 80–94. MR 871, DOI https://doi.org/10.1090/S0002-9947-1940-0000871-3
- Einar Hille, A class of differential operators of infinite order, I, Duke Math. J. 7 (1940), 458–495. MR 3228
- Einar Hille, Sur les fonctions analytiques définies par des séries d’Hermite, J. Math. Pures Appl. (9) 40 (1961), 335–342 (French). MR 143884 E. C. TITCHMARSH, The Theory of Functions, Oxford Univ. Press, London, 1939.
- Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708. MR 182834, DOI https://doi.org/10.2307/2373069
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Keywords:
Hermite function series
Article copyright:
© Copyright 1980
American Mathematical Society