Best uniform approximation to a class of rational functions

摘要

We explicitly determine the best uniform polynomial approximation p(n-1)*, to a class of rational functions of the form 1/(x - c)(2) + K (a, b, c, n)/(x - c) on [a, b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n - 1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle eta in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions. (c) 2006 Elsevier Inc. All rights reserved.