2017
Том 69
№ 6

All Issues

Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions

Shchitov A. N., Vakarchuk S. B.

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Abstract

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions $f(x) ∈ L_2^r(r ∈ ℤ_{+})$ and expressions containing moduli of continuity of the $k$th order $ω_k(f^{(r)}, t)$. Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from $L_2$. For the classes $F (k, r, Ψ*)$ defined by $ω_k(f^{(r)}, t)$ and the majorant $Ψ(t)=t^{4k/π^2}$, we determine the exact values of different widths in the space $L_2$.

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 11, pp 1738–1747.

Citation Example: Shchitov A. N., Vakarchuk S. B. Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions // Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1458-1466.

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