Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities

Article type: Research Article

Authors: Sidi, Avram

Affiliations: Computer Science Department, Technion – Israel Institute of Technology, Haifa 32000, Israel. E-mail: asidi@cs.technion.ac.il; URL: http://www.cs.technion.ac.il/~asidi/

Abstract: Let Σ∞n=0en[f]Pn(x) be the Legendre expansion of a function f(x) on (−1, 1). In this work, we derive an asymptotic expansion as n→∞ for en[f], assuming that f∈C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x=±1. Specifically, we assume that f(x) has asymptotic expansions of the forms  f(x)~Σ∞s=0Usl(log(1−x))(1−x)αs as x→1−,  f(x)~Σ∞s=0Vs(log(1+x))(1+x)βs as x→−1+, where Us(y) and Vs(y) are some polynomials in y. Here, αs and βs are in general complex and Rαs, Rβs>−1. An important special case is that in which Us(y) and Vs(y) are constant polynomials; for this case, the asymptotic expansion of en[f] assumes the form  en[f]~Σs=0αs∉Z+∞Σ∞i=0asihαs+i+1/2+(−1)n Σs=0βs∉Z+∞Σ∞i=0bsihβs+i+1/2 as n→∞, where h=(n+1/2)−2, Z+={0, 1 , 2, …}, and asi and bsi are constants independent of n.

Keywords: Legendre expansion, endpoint singularities, asymptotic expansions

DOI: 10.3233/ASY-2009-0950

Journal: Asymptotic Analysis, vol. 65, no. 3-4, pp. 175-190, 2009