Uniform rational approximation of |X|

Abstract

We review recent results about best uniform rational approximants of |x| on [−1, 1]. We shall sketch the proof of the theorem that Open image in new window where E nn (|x|,[−1, 1]) denotes the minimal approximation error. Related results are concerned with the asymptotic distribution of poles and zeros of the approximants r*n and the distribution of extreme points of the error function |x|−r*n(x) on [−1, 1]. Two conjectures about a generalisation of the approximation problem of |x| on [−1, 1] or of xα, α>0, on [0, 1] will be formulated.