Distribution of Alternation Points in Best Rational Approximations

A. I. Bogolyubskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study the convergence of counting measures of alternation point sets in best rational approximations to the equilibrium measure. It is shown that, for any prescribed nondecreasing sequence of denominator degrees, there exists a function analytic on $[0,1]$ and a sequence of numerator degrees such that the corresponding sequence of measures does not converge to the equilibrium measure of the interval.

Keywords: best rational approximation, alternation point, equilibrium measure, counting measure, Chebyshev (Walsh) table, Chebyshev rational operator.

UDC: 517.518.8

Received: 02.05.2007
Revised: 15.10.2007

DOI: 10.4213/mzm4570

 English version: Mathematical Notes, 2008, 83:4, 454–462

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