Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint
type: Item
instance of: scholarly article
title: Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint
zbMATH Open document ID: 0857.42015
DOI: 10.1016/0893-9659(96)00003-1
published in: Applied Mathematics Letters
publication date: 1997-03-12T00:00:00Z
review text: Let \(f(\varepsilon)\) be a divergent power series \(f(\varepsilon)\sim\sum^\infty_{j=0} b_n\varepsilon^n\) with \(b_n=o((n!)^{2G})\) where \(G\) is the ``Gevrey order'' of the asymptotic series. The two cases when \(G\) is infinite or zero are exemplified by functions \[ f_{\text{sub}}(\varepsilon)=\int^\infty_0 dt {\exp(-\exp t)\over (1+\varepsilon t)}\quad\text{and}\quad f_{\text{sup}}(\varepsilon;A)=\sqrt{{A\over \pi}} \int^\infty_0 {\exp(-A\log^2 t)\over 1+\varepsilon t} {dt\over t}. \] Asymptotic approximations for the coefficients of those functions are derived. They are limits with \(r\to0+\) and \(r\to1-\) of well-known asymptotics [see, for instance, \textit{G. Németh}, Mathematical approximation of special functions: ten papers on Chebyshev expansions (Nova Science, New York) (1992), \textit{J. P. Boyd}, Math. Comput. 39, 201-206 (1982; Zbl 0524.41014)]: \[ a^{\text{sub}}_n\sim(-\text{const }n/\log^{1/2}(n))\quad\text{and}\quad a^{\sup}_n=o(\exp(-\text{const }n^r))\quad\text{for any } r>0. \]
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 915067
zbMATH Keywords: Fourier series in Chebyshev polynomials
zbMATH Keywords: divergent power series
author: John P. Boyd
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
full work available at URL: 0893-9659(96)00003-1
OpenAlex ID: W2087738476
cites work: Q3202490
cites work: On the Convergence of the Chebyshev Series for Functions Possessing a Singularity in the Range of Representation
cites work: Asymptotic Estimates of Fourier Coefficients
cites work: Q4877137
cites work: The Rate of Convergence of Chebyshev Polynomials for Functions Which Have Asymptotic Power Series About One Endpoint
cites work: A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions
cites work: Stokes’s phenomenon for superfactorial asymptotic series
cites work: Q3207312
cites work: Q3219268
cites work: Q3993205
description: scientific article; zbMATH DE number 915067