RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (1805 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.5
MSC: 41A20

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} 

• http://mi.mathnet.ru/eng/izv1796
• http://mi.mathnet.ru/eng/izv/v41/i1/p182

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. A. Khatamov, “Inverse theorems in the theory of rational approximations of functions of several variables”, Math. Notes, 54:2 (1993), 858–866
•  Number of views: This page: 1236 Full text: 114 References: 30 First page: 1