On one estimate of divided differences and its applications
We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their gen...
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| Date: | 2019 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1434 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated
Whitney and Marchaud inequalities and their generalization to the Hermite interpolation.
For example, one of the numerous corollaries of this estimate is the fact that, given a function $f \in C(r)(I)$ and a set $Z = \{ z_j\}^{\mu}_{j=0}$ such that $z_{j+1} - z_j \geq \lambda | I|$ for all $0 \leq j \leq \mu 1$, where $I := [z_0, z_{\mu} ], | I|$ is the length of
$I$, and $\lambda$ is a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality
$\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$,
$$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1}
\int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$
where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $. |
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