This article is cited in
1 paper
Polynomial and rational approximation of functions of several variables with convex
derivatives in the $L_p$-metric $(0<p\leqslant\infty)$
A. Khatamov
Abstract:
Let
$\operatorname{Conv}_n^{(l)}(\mathscr G)$ be the set of all functions
$f$ such that for every
$n$-dimensional unit vector
$\mathbf e$ the
$l$th derivative in the direction of
$\mathbf e$,
$D^{(l)}(\mathbf e)f$, is continuous on a convex bounded domain
$\mathscr G\subset\mathbf R^n$ $(n\geqslant 2)$ and convex (upwards or downwards) on the nonempty intersection of every line
$L\subset\mathbf R^n$ with the domain
$\mathscr G$, and let $M^{(l)}(f,\mathscr G)\colon=\sup\{\|D^{(l)}(\mathbf e)f\|_{C(\mathscr G)}\colon\mathbf e\in \mathbf R^n$,
$\|\mathbf e\|=1\}<\infty$.
Sharp, in the sense of order of smallness, estimates of best simultaneous polynomial approximations of the functions
$f\in\operatorname{Conv}_n^{(l)}(\mathscr G)$ for which
$D^{(l)}(\mathbf e)f\in\operatorname{Lip}_K1$ for every
$\mathbf e$, and their derivatives in the metrics of
$L_p(\mathscr G)$ $(0<p\leqslant\infty)$ are obtained. It is proved that the corresponding parts of these estimates are preserved for best rational approximations, on any
$n$-dimensional parallelepiped
$Q$, of functions
$f\in\operatorname{Conv}_n^{(l)}(Q)$ in the metrics of
$L_p(Q)$ $(0<p<\infty)$ and it is shown that they are sharp in the sense of order of smallness for
$0<p\leqslant 1$.
UDC:
517.51
MSC: 41A10,
41A20,
41A63 Received: 15.10.1992
Fulltext:
PDF file (729 kB)