On approximation of ultraspherical polynomials in the oscillatory region

JOURNAL OF APPROXIMATION THEORY

卷:222

Article

Krasikov, Ilia¹ 

[1] Brunel Univ London, Dept Math, Uxbridge UB8 3PH, Middx, England

关键词: Orthogonal polynomials;    Ultraspherical polynomials;    Gegenbauer polynomials;    Uniform approximation;   

DOI  :  10.1016/j.jat.2017.07.003

来源: Elsevier

【 摘 要 】

For k >= 2 even, and a >= -(2k + 1)/4, we provide a uniform approximation of the ultraspherical polynomials Pk((alpha, alpha))(x) in the oscillatory region with a very explicit error term. In fact, our result coversall a for which the expression oscillatory region makes sense. To that end, we construct the almost ( equioscillating function g(x) = c root b(x) (1 - x(2))P-(alpha+1)/2(k)(alpha,alpha) (x) =cos B(x) +(x) Here the constantc = c(k, a) is defined by the normalization of Pk(alpha,alpha) (x), B(x) =integral(x)(0) b(x)dx, and the functions b(x) and B(x), as well as bounds on the error term r(x), are given by some rather simple elementary functions. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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