Geodesic
Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space~$L_2$ and Widths of Classes of Functions
Matematičeskie zametki, Tome 99 (2016) no. 2, pp. 215-238

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We obtain exact constants in Jackson-type inequalities for smoothness characteristics $\Lambda_k(f)$, $k\in \mathbb{N}$, defined by averaging the $k$th-order finite differences of functions $f \in L_2$. On the basis of this, for differentiable functions in the classes $L^r_2$, $r\in \mathbb{N}$, we refine the constants in Jackson-type inequalities containing the $k$th-order modulus of continuity $\omega_k$. For classes of functions defined by their smoothness characteristics $\Lambda_k(f)$ and majorants $\Phi$ satisfying a number of conditions, we calculate the exact values of certain $n$-widths.
Keywords: best polynomial approximation, smoothness characteristics, Jackson-type inequality, modulus of continuity, Bernstein $n$-width of a function class, Rolle's theorem. 
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     author = {S. B. Vakarchuk and V. I. Zabutnaya},
     title = {Inequalities between {Best} {Polynomial} {Approximations} and {Some} {Smoothness} {Characteristics} in the {Space~}$L_2$ and {Widths} of {Classes} of {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {215--238},
     publisher = {mathdoc},
     volume = {99},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_99_2_a5/}
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S. B. Vakarchuk; V. I. Zabutnaya. Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space~$L_2$ and Widths of Classes of Functions. Matematičeskie zametki, Tome 99 (2016) no. 2, pp. 215-238. http://geodesic.mathdoc.fr/item/MZM_2016_99_2_a5/