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This article is cited in 3 scientific papers (total in 3 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.
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”, Math. USSR-Izv., 11:1 (1977), 171–192
Linking options:
https://www.mathnet.ru/eng/im1796https://doi.org/10.1070/IM1977v011n01ABEH001698 https://www.mathnet.ru/eng/im/v41/i1/p182
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Abstract page: | 1406 | Russian version PDF: | 155 | English version PDF: | 13 | References: | 48 | First page: | 1 |
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