Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations

For a given strictly decreasing sequence $\{a_n\}^\infty_{n=0}$ of real numbers convergent to zero, we construct a continuous function $g$ on the closed interval $[-1,1]$ such that $R_{2n}(g)$ and $a_n$ have identical order of decrease as $n\to\infty$. Here $R_{n}(g)$ are the best approximations on the closed interval $[-1,1]$ in the uniform norm of the function $g$ by algebraic rational functions of degree at most $n$.