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

Let $f(z)=\int (z-x{)}^{-1}d\mu (x)$, where $\mu$ is a Borel measure supported on several subintervals of $(-1,1)$ with smooth Radon-Nikodym derivative. 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 rational approximant to $f(z)$ on the unit circle with $n$ poles inside the unit disk.