Geodesic
Remark on a Problem of Rational Approximation
Matematičeskie zametki, Tome 74 (2003) no. 3, pp. 446-448.

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We show that for any nonincreasing number sequence $\{a_n\}^{\infty}_{n=0}$ converging to zero, there exists a continuous $2\pi$-periodic function $g$ such that the sequence of its best uniform trigonometric rational approximations $\{R_n(g,C_{2\pi})\}^{\infty}_{n=0}$ and the sequence $\{a_n\}^{\infty}_{n=0}$ have the same order of decay. 
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A. P. Starovoitov. Remark on a Problem of Rational Approximation. Matematičeskie zametki, Tome 74 (2003) no. 3, pp. 446-448. http://geodesic.mathdoc.fr/item/MZM_2003_74_3_a12/

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