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The density of alternation points in rational approximation. (English) Zbl 0688.41018
The authors study the behaviour of the alternation (equioscillation) points for the error in best uniform rational approximation of an fC[-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 A. Kroo and 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 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.
Reviewer: G.D.Dikshit

MSC:
41A20Approximation by rational functions