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

Abstract. Let ∑∞ n=0 en[f ]Pn(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 → ∞ for en[f ], assuming that f ∈ C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x = ±1. In the present work, we extend this study to functions f(x) that are infinitely differentiable on [0, 1], except at finitely many points x1, . . . , xm in (−1, 1) and possibly at one or both of the endpoints x0 = 1 and xm+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