Sharp Estimates of Linear Approximations by Nonperiodic Splines in Terms of Linear Combinations of Moduli of Continuity
- 作者: Vinogradov O.1, Gladkaya A.1
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隶属关系:
- St. Petersburg State University
- 期: 卷 234, 编号 3 (2018)
- 页面: 303-317
- 栏目: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/241863
- DOI: https://doi.org/10.1007/s10958-018-4006-7
- ID: 241863
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详细
Assume that σ > 0, r, μ ???? ℕ, μ ≥ r + 1, r is odd, p ???? [1,+∞], and \( f\kern0.5em \in \kern0.5em {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). We construct linear operators Xσ,r,μ whose values are splines of degree μ and of minimal defect with knots \( \frac{k\pi}{\sigma },k\in \mathrm{\mathbb{Z}} \), such that
\( {\displaystyle \begin{array}{l}{\left\Vert f-{X}_{\sigma, r,u}(f)\right\Vert}_p\le {\left(\frac{\pi }{\sigma}\right)}^r\left\{\frac{A_r,0}{2}\left.{\upomega}_1\right|{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p+\sum \limits_{v=1}^{u-r-1}{A}_{r,v}{\omega}_v{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p\right\}\\ {}\kern9em +{\left(\frac{\pi }{\sigma}\right)}^r\left(\frac{{\mathcal{K}}_r}{\pi^r}-\sum \limits_{v=0}^{u-r-1}{2}^v{A}_{r,v}\right){2}^{r-\mu }{\omega}_{\mu -r}{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p,\end{array}} \) where for p = 1, . . . ,+∞, the constants cannot be reduced on the class \( {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). Here \( {\mathcal{K}}_r=\frac{4}{\pi}\sum \limits_{l=0}^{\infty}\frac{{\left(-1\right)}^{l\left(r+1\right)}}{{\left(2l+1\right)}^{r+1}} \) are the Favard constants, the constants Ar,ν are constructed explicitly, and ωv is a modulus of continuity of order ν. As a corollary, we get the sharp Jackson type inequality
作者简介
O. Vinogradov
St. Petersburg State University
编辑信件的主要联系方式.
Email: olvin@math.spbu.ru
俄罗斯联邦, St. Petersburg
A. Gladkaya
St. Petersburg State University
Email: olvin@math.spbu.ru
俄罗斯联邦, St. Petersburg