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On the asymptotic distribution of oscillation points in rational approximation. (English) Zbl 0791.41018
Let R m,n denote the set of algebraic rational functions of degree (m,n) on [-1,1]. For fC[-1,1] the distribution of the oscillation points of the best uniform rational approximation to f from R m,n is studied. For n=0, m, Kadec proved that one gets uniform distribution of the oscillatory points (with respect to Chebyshev measure). This result was shown recently by Borwein, Grothmann, Kroó and Saff to still hold if n=n(m) with n=o(m); whereas, for n=m-1, it was shown to no longer be true. In this paper, it is shown that Kadec’s result also holds for n=[cm] with 0<c<1, which had remained open in the previous paper.
MSC:
41A20Approximation by rational functions
41A50Best approximation, Chebyshev systems
References:
[1]N. I. Achieser,Vorlesungen über Approximationstheorie, Akademie Verlag (Berlin, 1953).
[2]P. B. Borwein, Approximations with negative roots and poles,J. Approx. Theory,35 (1982), 132–141. · Zbl 0487.41017 · doi:10.1016/0021-9045(82)90031-4
[3]P. B. Borwein, R. Grothmann, A. Kroó, andE. B., Saff, The density of alternation points in rational approximation,Proc. Amer. Math. Soc.,106 (1989), 881–888. · doi:10.1090/S0002-9939-1989-0948147-0
[4]M. I. Kadec, On the distribution of points of maximal deviation in the approximation of continuous functions by polynomials,Uspekhi Mat. Nauk,15 (1960), 199–202. [In Russian].
[5]A. Kroó and.F. Peherstorfer, Interpolator properties of best rational L1-approximations,Constr. Approx.,4 (1988), 97–106. · Zbl 0676.41018 · doi:10.1007/BF02075450
[6]A. Kroó andE. B. Saff, The density of extreme points in complex polynomial approximation,Proc. Amer. Math. Soc.,103 (1988), 203–209.
[7]G. G. Lorentz, Distribution of alternation points in uniform polynomial approximation,Proc. Amer. Math. Soc. 92 (1984), 401–403. · doi:10.1090/S0002-9939-1984-0759662-2
[8]H. Werner, On the rational Tschebyscheff operator,Math. Z.,86 (1964), 317–326. · Zbl 0206.07504 · doi:10.1007/BF01110406