Computational Methods and Function Theory 3 (2003), No. 1, 165--177
Copyright Heldermann Verlag 2003
Hans-Peter Blatt
mga009@ku-eichstaett.de, Katholische Universität Eichstätt, Mathematisch-Geographische Fakultät, 85071 Eichstätt, Germany.
René Grothmann
rene.grothmann@ku-eichstaett.de, Katholische Universität Eichstätt, Mathematisch-Geographische Fakultät, 85071 Eichstätt, Germany.
Ralitza Kovacheva
rkovach@math.bas.bg, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.
The distribution of equi-oscillation points (alternation points) for the error in best Chebyshev approximation on [-1, 1] by rational functions is investigated. In general, the alternation points need not be dense in [-1, 1] when rational functions of degree (n, m) are considered and asymptotically n/m ® κ with κ ≥ 1. We show that the asymptotic behavior of the alternation points is closely related to the behavior of the poles of the rational approximants. Hence, poles of the rational approximations are attracting points of alternations such that the well-known equi-distribution for the polynomial case can be heavily disturbed.
Keywords: Rational approximation, alternation points.
MSC 2000: Primary 41A20.
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