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Let $ f(z)=\int(z-x)^{-1}d\mu(x) $, where $ \mu $ is a Borel measuresupported on several subintervals of $ (-1,1) $ with smooth Radon-Nikodymderivative. We study strong asymptotic behavior of the error of approximation $(f-r_n)(z) $, where $ r_n(z) $ is the $ L_{\mathbb R}^2$-best rationalapproximant to $ f(z) $ on the unit circle with $ n $ poles inside the unitdisk.

On $L_{\mathbb R}^2$-best rational approximants to Markov functions on several intervals