Approximation by Simplest Fractions

In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions $f(z)$ on compact subsets of the complex plane by simplest fractions of the form $\Theta _n(z)=\sum _{j=1}^n1/(z-z_j)$ are considered. In particular, it is shown that the best approximation of a function $f$ by the fractions $\Theta _n$ is of the same order of vanishing as the best approximations by polynomials of degree $\le n$.