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Zapiski Nauchnykh Seminarov POMI, 1994, Volume 217, Pages 92–111
(Mi znsl5963)
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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
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
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
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https://www.mathnet.ru/eng/znsl5963 https://www.mathnet.ru/eng/znsl/v217/p92
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