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Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities

Author: Avram Sidi
Journal: Math. Comp. 80 (2011), 1663-1684
MSC (2000): Primary 40A05, 40A10, 41A58, 41A60, 42C10
Published electronically: December 30, 2010
MathSciNet review: 2785473
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Abstract: Let $ \sum^\infty_{n=0}e_n[f]P_n(x)$ be the Legendre expansion of a function $ f(x)$ on $ (-1,1)$. In an earlier work [A. Sidi, Asymptot. Anal., 65 (2009), pp. 175-190], we derived asymptotic expansions as $ n\to\infty$ for $ e_n[f]$, assuming that $ f\in C^\infty(-1,1)$, but may have arbitrary algebraic-logarithmic singularities at one or both endpoints $ x=\pm1$. In the present work, we extend this study to functions $ f(x)$ that are infinitely differentiable on $ [0,1]$, except at finitely many points $ x_1,\ldots,x_m$ in $ (-1,1)$ and possibly at one or both of the endpoints $ x_0=1$ and $ x_{m+1}=-1$, where they may have arbitrary algebraic singularities, including finite jump discontinuities. Specifically, we assume that, for each $ r$, $ f(x)$ has asymptotic expansions of the form

$\displaystyle f(x)\sim\sum^\infty_{s=0}W^{(\pm)}_{rs}\vert x-x_r\vert^{\delta^{(\pm)}_{rs}}$   as $x&rarr#to;x_r±$,

where $ W^{(\pm)}_{rs}$ and $ \delta^{(\pm)}_{rs}$ are, in general, complex and $ \Re\delta^{(\pm)}_{rs}>-1$. We derive the full asymptotic expansion of $ e_n[f]$ as $ n\to\infty$ for this very general behavior of $ f(x)$. In the special case where $ \delta^{(\pm)}_{rs}=\sigma^{(\pm)}_{r}+s$, $ 1\leq r\leq m$, and $ \delta^{(-)}_{0s}=\alpha+s$ and $ \delta^{(+)}_{m+1,s}=\beta+s$, this expansion reduces to

$\displaystyle e_n[f]\sim \sum^{m}_{r=1} \bigg\{ e^{\mrm{i}\widehat{n}\theta_r}$ $\displaystyle \bigg[\sum^\infty_{s=0} \frac{a^{(+)}_{rs}} {\widehat{n}^{\sigma^... ...m^\infty_{s=0} \frac{a^{(-)}_{rs}} {\widehat{n}^{\sigma^{(-)}_{r}+s+1/2}}\bigg]$    
$\displaystyle +e^{-\mrm{i}\widehat{n}\theta_r}$ $\displaystyle \bigg[\sum^\infty_{s=0} \frac{\widehat{a}^{(+)}_{rs}} {\widehat{n...{\widehat{a}^{(-)}_{rs}} {\widehat{n}^{\sigma^{(-)}_{r}+s+1/2}}\bigg] \bigg\}$    
$\displaystyle +\sum^\infty_{\substack{s=0 \\ \alpha\not\in \mathbb{Z}^+}}$ $\displaystyle \frac{A_s}{\widehat{n}^{2(\alpha+s+1/2)}} +(-1)^n \sum^\infty_{\s... ...ack{s=0 \\ \beta\not\in\mathbb{Z}^+}} \frac{B_s} {\widehat{n}^{2(\beta+s+1/2)}}$   as $ n\to\infty$.    

where $ \theta_r=\cos^{-1}x_r$, $ \widehat{n}=n+1/2$, $ \mathbb{Z}^+=\{0,1,2,\ldots\},$ and $ a^{(\pm)}_{rs}$, $ \widehat{a}^{(\pm)}_{rs}$, $ A_s$, and $ B_s$ are constants independent of $ n$. In the course of this study, we also derive a full asymptotic expansion as $ n\to\infty$ for integrals of the form $ \int^d_cf(x)P_n(x) dx$ where $ [c,d]\in(-1,1)$ and $ f\in C^\infty[c,d]$ or $ f\in C^\infty(c,d)$ but may have arbitrary algebraic singularities at $ x=c$ and/or $ x=d$.

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Additional Information

Avram Sidi
Affiliation: Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Keywords: Legendre polynomials, Legendre series, interior singularities, endpoint singularities, asymptotic expansions.
Received by editor(s): March 17, 2010
Received by editor(s) in revised form: May 25, 2010
Published electronically: December 30, 2010
Additional Notes: This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2008399.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.