Sign changes in rational L w 1 -approximation

Let f ∈ L _w ¹ [−1, 1], let r _n,m(f) be the best rational L _w ¹ -approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim _{n → ∞} (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r _n,m (f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006.