Polynomial approximation over two intervals

Question

I want to approximate the function $f(x) = \frac{1}{x - z}$, $z \in \mathbb{C}$, on two intervals $[a,b] \cup [c,d] \subset\mathbb{R}$ using polynomials. If $z$ was real and $b = c$, Bernstein's theorem on polynomial approximation of analytic functions would immediately yield the convergence rate. Is there an extension which would cover my case?

Answer 1

This is an application of logarithmic potential theory. See e.g. "Logarithmic Potential Theory with Applications to Approximation Theory" by Edward Saff for an introduction to logarithmic potential theory and its relation to polynomial approximation, and "The Potential Theory of Several Intervals and Its Applications" by Shen et al. for the Green's function for several intervals on the real axis.