The concept of alternation sets is well known in the theory of best polynomial approximation and the extension to real rational approximation has been studied extensively.
Let be real valued and let denote the set of rational functions of the form , where and and are polynomials of degrees at most and respectively. For each pair of nonnegative integers there then exists a unique function that satsfies
where is the sup-norm on . Write , with numerator and denominator having no common factors, and define the defect by
Then there exist points on the closed interval
where , fixed; this is called an alternation set (in general not unique). For the sequel, let for each pair an arbitrary, fixed alternation set for the best approximation to be given by
In the paper of P. B. Borwein, A. Króo, R. Grothmann, E. B. Saff, Proc. Am. Math. Soc. 105, No.4, 881-888 (1989; Zbl 0688.41018)], it is shown that a subsequence of alternation sets is dense in whenever . In “Behavior of alternation points in best rational approximation”, [Acta Appl. Math. 33, No.2-3, 195-210 (1993; Zbl 0801.41021)], the connection between denseness and the number of poles of that lie outside an -neighborhood of is studied.
In the paper under review the main result now is a new theorem that implies (amongst others) many of the results found in the papers quoted above. Let be a nonrational function and let satisfy
Then there exists a subsequence such that
in the weak topology; here
the normalized counting measure. Furthermore if are the zeros of the polynomial of degree , then
(the normalized counting measure of all poles of ), and the function is introduced using the logarithmic potential of .
A clearly written and interesting paper.