Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions
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 res...
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| Date: | 2004 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2004
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3858 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 $L_2$. |
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