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Zapiski Nauchnykh Seminarov POMI, 1994, Volume 217, Pages 92–111 (Mi znsl5963)  

This article is cited in 1 scientific paper (total in 1 paper)

Spectral synthesis in the Sobolev space associated with integral metric

Yu. V. Netrusov

St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences
Full-text PDF (753 kB) Citations (1)
Abstract: The aim of this paper is to prove Theorem A.
Theorem A. Let $l\in\mathbb N$, $A\subset\mathbb R^n$. Then the following two conditions are equivalent:
1) for any $\varepsilon>0$ there exist a function $f_\varepsilon$ and an open set $G\supset A$ such that

$$ \operatorname{supp}f_\varepsilon\subset\mathbb R^n\setminus G,\qquad\|f-f_\varepsilon\|_{W^l_1}\le\varepsilon; $$

2) for any $\alpha=(\alpha_1,\dots,\alpha_n)\in\{0,1,2,\dots,\}^n$, $|\alpha|=\alpha_1+\dots+\alpha_n<l$, there exists a set $E_\alpha$ with the following properties:
a) if $n\le l-|\alpha|$ then $E_\alpha=A$;
b) if $n>l-|\alpha|$ then the Hausdorff measure of order $n-l+|\alpha|$ of set $A\setminus E_\alpha$ is equal to zero;
c) for any point $x\in E_\alpha$ the following relation holds:

$$ \lim_{a\to0}a^{-n}\int_{D(x,a)}|D^\alpha f(y)|\,dy=0, $$
where $D(x,a)$ is the ball of radius $a>0$ centered at $x\in\mathbb R^n$.
Some generalizations of this result are also proved. Bibliography: 9 titles.
Received: 15.02.1994
English version:
Journal of Mathematical Sciences (New York), 1997, Volume 85, Issue 2, Pages 1814–1826
DOI: https://doi.org/10.1007/BF02355292
Bibliographic databases:
Document Type: Article
UDC: 517.51
Language: Russian
Citation: Yu. V. Netrusov, “Spectral synthesis in the Sobolev space associated with integral metric”, Investigations on linear operators and function theory. Part 22, Zap. Nauchn. Sem. POMI, 217, POMI, St. Petersburg, 1994, 92–111; J. Math. Sci. (New York), 85:2 (1997), 1814–1826
Citation in format AMSBIB
\Bibitem{Net94}
\by Yu.~V.~Netrusov
\paper Spectral synthesis in the Sobolev space associated with integral metric
\inbook Investigations on linear operators and function theory. Part~22
\serial Zap. Nauchn. Sem. POMI
\yr 1994
\vol 217
\pages 92--111
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5963}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1327518}
\zmath{https://zbmath.org/?q=an:0882.46016|0907.46029}
\transl
\jour J. Math. Sci. (New York)
\yr 1997
\vol 85
\issue 2
\pages 1814--1826
\crossref{https://doi.org/10.1007/BF02355292}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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