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 $n$. 
@article{MZM_2001_69_6_a11,
     author = {A. P. Starovoitov},
     title = {On the {Problem} of {Describing} {Sequences} of {Best} {Trigonometric} {Rational} {Approximations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {919--924},
     publisher = {mathdoc},
     volume = {69},
     number = {6},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_6_a11/}
}
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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/