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 $n$. 
@article{MZM_2003_74_5_a10,
     author = {A. P. Starovoitov},
     title = {Existence of {Continuous} {Functions} with a {Given} {Order} of {Decrease} of {Least} {Deviations} from {Rational} {Approximations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {745--751},
     publisher = {mathdoc},
     volume = {74},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/}
}
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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/