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This article is cited in 1 scientific paper (total in 1 paper)
Approximations by convolutions and antiderivatives
A. M. Sedletskii M. V. Lomonosov Moscow State University
Abstract:
Let $g$ be a given function in $L^1=L^1(0,1)$, and let $B$ be one of the spaces $L^p(0,1)$, $1\le p<\infty$, or $C_0[0,1]$. We prove that the set of all convolutions $f*g$, $f\in B$, is dense in $B$ if and only if $g$ is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on $g$, we prove the equivalence in $B$ of the systems $f_n*g$ and $If_n$, where $f_n\in L^1$, $n\in\mathbb N$, and $If=f*1$ is the antiderivative of $f$. As a consequence, we obtain criteria for the completeness and basis property in $B$ of subsystems of antiderivatives of $g$.
Received: 29.12.2004
Citation:
A. M. Sedletskii, “Approximations by convolutions and antiderivatives”, Mat. Zametki, 79:5 (2006), 756–766; Math. Notes, 79:5 (2006), 697–706
Linking options:
https://www.mathnet.ru/eng/mzm2747https://doi.org/10.4213/mzm2747 https://www.mathnet.ru/eng/mzm/v79/i5/p756
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Abstract page: | 475 | Full-text PDF : | 257 | References: | 56 | First page: | 3 |
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