Uniform rational approximations of functions of class $V_r$

Let $V_r$ denote the set of functions $f$, defined on a finite interval $[a,b]$, for which $f^{(r-1)}$ is absolutely continuous on $[a,b]$ and is a primitive of a function of bounded variation; let $R_n(f)$ denote the best uniform approximation of $f$ by rational functions of order $n$. It is shown that $R_n(f)=o(n^{-r-1})$ for every $f\in V_r$ $(r\geqslant1)$, and that this estimate is of best possible order for the class $V_r$.
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