Approximation by simple partial fractions and sums of shifts of a single function

Korevaar's theorem states that simple partial fractions $\sum\limits_{k=1}^n \frac{1}{z - w_k}$ with poles on the boundary of a bounded simply connected domain $D$ are dense in the space of analytic functions on $D$ in the uniform convergence topology. We will discuss various generalizations of Korevaar's theorem based on general results on the density of quantized approximations in Hilbert space. In particular, we will talk about uniform approximation of harmonic functions by the sum of shifts of a single function.