Best uniform approximation to a class of rational functions

Abstract : 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 η 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.
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Journal of Mathematical Analysis and applications, Elsevier, 2007, 334 (2), pp.909-921. 〈10.1016/j.jmaa.2006.10.047〉
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Zhitong Zheng, Jun-Hai Yong. Best uniform approximation to a class of rational functions. Journal of Mathematical Analysis and applications, Elsevier, 2007, 334 (2), pp.909-921. 〈10.1016/j.jmaa.2006.10.047〉. 〈inria-00517597〉

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