Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities

Abstract

Let Σ^∞_n=0e_n[f]P_n(x) be the Legendre expansion of a function f(x) on (−1, 1). In this work, we derive an asymptotic expansion as n→∞ for e_n[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=0U_sl(log(1−x))(1−x)^α_s as x→1−,

 f(x)~Σ^∞_s=0V_s(log(1+x))(1+x)^β_s as x→−1+,

where U_s(y) and V_s(y) are some polynomials in y. Here, α_s and β_s are in general complex and Rα_s, Rβ_s>−1. An important special case is that in which U_s(y) and V_s(y) are constant polynomials; for this case, the asymptotic expansion of e_n[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 a_si and b_si are constants independent of n.