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)
The interaction of alternation points and poles in rational approximation. (English) Zbl 1072.41005
The author investigates the interrelation of alternation points for the minimal error function and poles of best Chebyshev approximants if uniform approximation on the interval $[-1, 1]$ by rational functions of degree $(n(s), m(s))$ is considered, $s \in N$. He shows that at least for a subsequence $\Lambda \subset N$, the asymptotic behaviour of the alternation points of the degree $(n(s), m(s))$, $s \in\Lambda$ is completely determined by the location of the poles of the best approximants and vice versa, if $m(s) \leq n(s)$ or $m(s) - n(s) = o(s/\log s)$ as $s \to\infty$.

MSC:
 41A20 Approximation by rational functions
Keywords:
Rational approximation
Full Text:
References:
 [1] Andrievskii, V. V.; Blatt, H. -P.: Discrepancy of signed measures and polynomial approximation. (2002) · Zbl 0995.30001 [2] Blatt, H. -P.; Grothmann, R.; Kovacheva, R. K.: Discrepancy estimates and rational Chebyshev approximation. Constructive theory of functions, 213-218 (2003) · Zbl 1028.41013 [3] Blatt, H. -P.; Grothmann, R.; Kovacheva, R. K.: Poles and alternation points in real rational Chebyshev approximation. Comput. methods funct. Theory 3, 165-177 (2003) · Zbl 1057.41008 [4] Borwein, P. B.; Kroó, A.; Grothmann, R.; Saff, E. B.: The density of alternation points in rational approximation. Proc. amer. Math. soc. 105, 881-888 (1989) · Zbl 0688.41018 [5] Braess, D.; Lubinsky, D. S.; Saff, E. B.: Behavior of alternation points in best rational approximation. Acta applicandae mathematicae 33, 195-210 (1993) · Zbl 0801.41021 [6] Kadec, M. I.: On the distribution of points of maximal deviation in the approximation of continuous functions by polynomials. Uspekhi mat. Nauk 15, 199-202 (1960) [7] Kroó, A.; Peherstorfer, F.: On the asymptotic distribution of oscillation points in rational approximation. Analysis Mathematica 19, 225-232 (1993) · Zbl 0791.41018 [8] Saff, E. B.; Stahl, H.: Asymptotic distribution of poles and zeros of best rational approximants to x$\alpha$ on [0,1]. Proceedings of the semester in complex analysis, 329-348 (1995) · Zbl 0826.41018 [9] Saff, E. B.; Totik, V.: Logarithmic potentials with external fields. (1997) · Zbl 0881.31001 [10] Saff, E. B.; Stahl, H.: Ray sequences of best rational approximants for $\vert x\vert \alpha$. Can. J. Math. 49, No. 5, 1034-1065 (1997) · Zbl 0893.41011