Estimates of Lebesgue constants for Lagrange interpolation processes by rational functions under mild restrictions to their fixed poles.

We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have accumulation points on the intervals. In particular, for arbitrary finite set on the compact $E$ which consists of several intervals of the real axis it is possible to find poles with accumulation points at the chosen finite set, but with the order of Lebesgue constants $\ln n$ on $E$ (what is usually is considered as optimal). To prove it we use an analogue of the inverse polynomial image method for rational functions with fixed poles.
(The talk is based on joint work with Sergei Kalmykov)