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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The density of alternation points in rational approximation

Authors: P. B. Borwein, A. Kroó, R. Grothmann and E. B. Saff
Journal: Proc. Amer. Math. Soc. 105 (1989), 881-888
MSC: Primary 41A20; Secondary 41A50
MathSciNet review: 948147
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Abstract: We investigate the behavior of the equioscillation (alternation) points for the error in best uniform rational approximation on $ [-1,1]$. In the context of the Walsh table (in which the best rational approximant with numerator degree $ \leq m$, denominator degree $ \leq n$, is displayed in the $ n$th row and the $ m$th column), we show that these points are dense in $ [-1,1]$, if one goes down the table along a ray above the main diagonal $ \left( {n = \left[ {cm} \right],c < 1} \right)$. A counterexample is provided showing that this may not be true for a subdiagonal of the table. In addition, a Kadec-type result on the distribution of the equioscillation points is obtained for asymptotically horizontal paths in the Walsh table.

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Keywords: Rational approximation, extreme points, best approximants
Article copyright: © Copyright 1989 American Mathematical Society