Article ID Journal Published Year Pages File Type
5773731 Journal of Approximation Theory 2017 14 Pages PDF
Abstract

For k≥2 even, and α≥−(2k+1)∕4, we provide a uniform approximation of the ultraspherical polynomials Pk(α,α)(x) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g(x)=cb(x)(1−x2)(α+1)∕2Pk(α,α)(x)=cosB(x)+r(x). Here the constant c=c(k,α) is defined by the normalization of Pk(α,α)(x), B(x)=∫0xb(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.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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