Approximation by simple partial fractions and the Hilbert transform

We study the problem of approximation of functions in $L_p$ by simple partial fractions on the real axis and semi-axis. A simple partial fraction is a rational function of the form $g(t)=\sum_{k=1}^n\frac1{t-z_k}$, where $z_1,\dots,z_n$ are complex numbers. We describe the set of functions that can be approximated by simple partial fractions within any accuracy and the set of functions that can be approximated by convex combinations of them (the cone of simple partial fractions). We obtain estimates for the norms of simple partial fractions and conditions for the convergence of function series $\sum_{k=1}^\infty\frac1{t-z_k}$ in the space $L_p$. Our approach is based on the use of the Hilbert transform and the methods of convex analysis.