Geodesic
Asymptotic behavior of best approximations of continuous functions
Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 551-569

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In this paper it is proved that in each class $W^rH_\omega$, $r=0,1,\dots$, with convex modulus of continuity $\omega$ there exists a function $f$ whose best approximations admit the limit behavior $$ \lim_{n\to\infty}E_n(f)/E_n(W^rH_\omega)=1. $$ Bibliography: 11 titles. 
@article{IM2_1977_11_3_a4,
     author = {V. N. Temlyakov},
     title = {Asymptotic behavior of best approximations of continuous functions},
     journal = {Izvestiya. Mathematics },
     pages = {551--569},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {1977},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1977_11_3_a4/}
}
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V. N. Temlyakov. Asymptotic behavior of best approximations of continuous functions. Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 551-569. http://geodesic.mathdoc.fr/item/IM2_1977_11_3_a4/