Geodesic
Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations
Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 745-751.

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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 $n$. 
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A. P. Starovoitov. Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations. Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 745-751. http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/

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