Geodesic
On the Problem of Describing Sequences of Best Trigonometric Rational Approximations
Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 919-924.

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For a strictly decreasing sequence $\{a_n\}^\infty_{n=0}$ of nonnegative real numbers converging to zero, we construct a continuous $2\pi$-periodic function $f$ such that $R^T_n(f)=a_n$, $n=0,1,2,\dots$, where $R^T_n(f)$ are best approximations of the function $f$ in uniform norm by trigonometric rational functions of degree at most $n$. 
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A. P. Starovoitov. On the Problem of Describing Sequences of Best Trigonometric Rational Approximations. Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 919-924. http://geodesic.mathdoc.fr/item/MZM_2001_69_6_a11/

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