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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 1, Pages 182–202 (Mi izv1796)  

This article is cited in 2 scientific papers (total in 2 papers)

Dependence of the differentiability of functions of several variables on their rate of approximation by rational functions

E. P. Dolzhenko, V. I. Danchenko


Abstract: Let $E$ be a Lebesgue measurable subset of a $k$-dimensional cube ($k\geqslant1$), let $f\in L_p[E]$, where $0<p\leqslant\infty$, and let $R_n[f,p,E]$ be the least deviation of $f$, in the metric of $L_p[E]$, from the rational functions of degre $\leqslant n$. If $R_n[f,p,E]=O(n^{-\lambda})$, then, for $0<\mu<\lambda$, $f$ has a local differential of order $\mu$ in the $L_p$-metric at each point $\xi\in E$, except perhaps points $\xi$ of some set of metric dimension $\leqslant k-1+(p\mu+1)/(p\lambda+1)$ (this inequality is sharp). In addition, $f$ has a global differential of order $\mu$ in the metric of $L_q [E]$ for any $q<p/(p\mu+1)$.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:1, 171–192

Bibliographic databases:

UDC: 517.5
MSC: 41A20
Received: 20.04.1976

Citation: E. P. Dolzhenko, V. I. Danchenko, “Dependence of the differentiability of functions of several variables on their rate of approximation by rational functions”, Izv. Akad. Nauk SSSR Ser. Mat., 41:1 (1977), 182–202; Math. USSR-Izv., 11:1 (1977), 171–192

Citation in format AMSBIB
\Bibitem{DolDan77}
\by E.~P.~Dolzhenko, V.~I.~Danchenko
\paper Dependence of the differentiability of functions of several variables on their rate of approximation by rational functions
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 1
\pages 182--202
\mathnet{http://mi.mathnet.ru/izv1796}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=442550}
\zmath{https://zbmath.org/?q=an:0355.41020|0392.41007}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 1
\pages 171--192
\crossref{https://doi.org/10.1070/IM1977v011n01ABEH001698}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. A. Sevast'yanov, “On an estimate for the smallness of sets of points of nondifferentiability of functions as related to the degree of approximation by rational functions”, Math. USSR-Izv., 26:2 (1986), 347–369  mathnet  crossref  mathscinet  zmath
    2. A. Khatamov, “Inverse theorems in the theory of rational approximations of functions of several variables”, Math. Notes, 54:2 (1993), 858–866  mathnet  crossref  mathscinet  zmath  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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