Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities

Avram Sidi Computer Science Department, Technion – Israel Institute of Technology, Haifa 32000, Israel E-mail: [email protected]; URL: http://www.cs.technion.ac.il/~asidi/ Abstract. Let ∑∞ n=0 en[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