Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities
Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities
复制标题
具有内部和端点奇点的函数的勒让德级数系数的渐近展开
作者：
A. Sidi
抽象的。设∑∞n=0 En[f]Pn(X)是函数f(X)在(−1，1)上的勒让德展开式。在较早的工作中[A.Sidi，渐近。，65(2009年)，第175-190页]，我们得到了关于En[f]的渐近展开式为n→∞，假设f∈C∞(−1，1)，但在一个或两个端点x=±1处可能有任意的代数-对数奇点。在本文中，我们将这一研究推广到在[0，1]上无限可微的函数f(X)，除了在有限多个点x1，.。。，Xm在(−1，1)中，并且可能在端点X0=1和Xm+1=−1中的一个或两个处，其中它们可以具有任意的代数奇点，包括有限跳跃间断。具体地说，我们假设，对于每个r，f(X)都有如下形式的渐近展开
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