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Zbl 0688.41018
Borwein, P.B.; Króo, A.; Grothmann, R.; Saff, E.B.
The density of alternation points in rational approximation. (English)
[J] Proc. Am. Math. Soc. 105, No.4, 881-888 (1989). ISSN 0002-9939; ISSN 1088-6826/e

The authors study the behaviour of the alternation (equioscillation) points for the error in best uniform rational approximation of an $f\in C[-1,1]$. The theorems proved here may be compared with some known results for best polynomial approximation given by some of these authors and some others [see {\it A. Kroo} and {\it E. B. Saff}, Proc. Am. Math. Soc. 103, No.1, 203-209 (1988; Zbl 0663.41027)]. They give three theorems. From the first theorem, in the context of Walsh table they deduce that these points are dense in the interval [-1,1] if one goes down the table along a ray above the main diagonal. In Theorem 2 they provide a result similar to one due to {\it M. I. Kadec} [Usp. Mat. Nauk 15, No.1(91), 199-202 (1960; Zbl 0136.364)] on polynomial approximations. In the third theorem they furnish a counter example to show that the result may not be true for a subdiagonal of the table.
[G.D.Dikshit]

MSC 2000:: *41A20 Approximation by rational functions

Keywords: alternation points; equioscillation points; error in best uniform rational approximation; Walsh table

Citations: Zbl 0663.41027; Zbl 0136.364

Cited in: Zbl 1165.41004 Zbl 1057.41008

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