A method of approximation by rational functions on the real line

For a given system of numbers $\{z_k\}_{k=1}^n$, $\operatorname{Im}z_k>0$, rational functions of order $4n-2$ are constructed which effect for a function $f(x)\in C_\infty$ an approximation of the same order as the best approximation by proper rational functions having poles at the points $\{z_k\}_{k=1}^n$ and $\{\overline z_k\}_{k=1}^n$.