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Russian Academy of Sciences. Izvestiya Mathematics, 1995, Volume 44, Issue 1, Pages 165–179
DOI: https://doi.org/10.1070/IM1995v044n01ABEH001587
(Mi im820)
 

This article is cited in 1 scientific paper (total 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
References:
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 $0<p\leqslant 1$.
Received: 15.10.1992
Bibliographic databases:
UDC: 517.51
MSC: 41A10, 41A20, 41A63
Language: English
Original paper language: Russian
Citation: A. Khatamov, “Polynomial and rational approximation of functions of several variables with convex derivatives in the $L_p$-metric $(0<p\leqslant\infty)$”, Russian Acad. Sci. Izv. Math., 44:1 (1995), 165–179
Citation in format AMSBIB
\Bibitem{Kha94}
\by A.~Khatamov
\paper Polynomial and rational approximation of functions of several variables with convex
derivatives in the $L_p$-metric $(0<p\leqslant\infty)$
\jour Russian Acad. Sci. Izv. Math.
\yr 1995
\vol 44
\issue 1
\pages 165--179
\mathnet{http://mi.mathnet.ru//eng/im820}
\crossref{https://doi.org/10.1070/IM1995v044n01ABEH001587}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1271519}
\zmath{https://zbmath.org/?q=an:0829.41018}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1995IzMat..44..165K}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1995QU91700008}
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  • https://www.mathnet.ru/eng/im/v58/i1/p167
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:286
    Russian version PDF:124
    English version PDF:15
    References:47
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