Geodesic
Jackson-Type Inequalities and Widths of Function Classes in~$L_2$
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 11-19

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The sharp Jackson-type inequalities obtained by Taikov in the space $L_2$ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of $k$th order $(k=2,3,\dots)$. We also obtain exact values of the $n$-widths of the function classes $F(k,r,\Phi)$ and $\mathcal{F}_k^r (h)$, which are a generalization of the classes $F(1,r,\Phi)$ and $\mathcal{F}^r_1(h)$ studied by Taikov.
Keywords: Jackson-type inequalities, width of function classes, modulus of continuity of $k$th order, periodic function, Bernstein, Kolmogorov, Gelfand $n$-widths. 
@article{MZM_2006_80_1_a1,
     author = {S. B. Vakarchuk},
     title = {Jackson-Type {Inequalities} and {Widths} of {Function} {Classes} in~$L_2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {11--19},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a1/}
}
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S. B. Vakarchuk. Jackson-Type Inequalities and Widths of Function Classes in~$L_2$. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 11-19. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a1/