# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\left(m,n\right)$ on $\left[-1,1\right]$. For $f\in C\left[-1,1\right]$ the distribution of the oscillation points of the best uniform rational approximation to $f$ from ${R}_{m,n}$ is studied. For $n=0$, $m\to \infty$, 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 $n=n\left(m\right)$ with $n=o\left(m\right)$; 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 $n=\left[cm\right]$ with $0, which had remained open in the previous paper.
##### MSC:
 41A20 Approximation by rational functions 41A50 Best approximation, Chebyshev systems
##### Keywords:
best uniform rational approximation
##### 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