The density of alternation points in rational approximation

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.