Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions

For a given nonincreasing vanishing sequence $\{a_n\}^\infty_{n=0}$ of nonnegative real numbers, we find necessary and sufficient conditions for a sequence $\{n_k\}^\infty_{k=0}$ to have the property that for this sequence there exists a function f continuous on the interval $[0,1]$ and satisfying the condition that $R_{n_k,m_k}(f)=E_{n_k}(f)=a_{n_k}$, $k=0,1,2,\dots$, where $E_n(f)$ and $R_{n,m}(f)$ are the best uniform approximations to the function $f$ by polynomials whose degree does not exceed $n$ and by rational functions of the form $r_{n,m}(x)=p_n(x)/q_m(x)$, respectively.