Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5773731 | Journal of Approximation Theory | 2017 | 14 Pages |
Abstract
For kâ¥2 even, and αâ¥â(2k+1)â4, 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)(1âx2)(α+1)â2Pk(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)=â«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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Ilia Krasikov,