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Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let $r_n\left(z\right)$ denote the rational function of degree n with poles at the points ${b_{ni}}_{i=1}^n$ and interpolating ⨍ at the points ${a_{ni}}_{i=0}^n$. We investigate how these points should be chosen to guarantee the convergence of $r_n$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles ${b_{ni}}_{i,n}$ without limit points on K. In this paper we study the case of general compact sets K, when such a separation...