The density of alternation points in rational approximation

Borwein, P. B.; Kroó, A.; Grothmann, R.; Saff, E. B.

doi:10.1090/S0002-9939-1989-0948147-0

The density of alternation points in rational approximation
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by P. B. Borwein, A. Kroó, R. Grothmann and E. B. Saff PDF

Proc. Amer. Math. Soc. 105 (1989), 881-888 Request permission

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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