On approximation of ultraspherical polynomials in the oscillatory region

Abstract

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