Approximation by linear fractional transformations of simple partial fractions and their differences

We study applications of a property of simple partial fractions such that a difference $f-\rho$, where $\rho$ is a simple partial fraction of order at most $n$, under linear-fractional transformations becomes again a difference of certain function and certain simple partial fraction of order at most $n$ with quadratic weight. We prove a theorem of uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates of best uniform approximation of certain functions on real semi-axis $\mathbb{R}^+$. For the first time, for continuous functions of rather common type we obtain estimates of best approximation by differences of simple partial fractions on $\mathbb{R}^+$, and for odd functions on all axis $\mathbb{R}$.