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Mat. Zametki, 2006, Volume 79, Issue 5, Pages 756–766 (Mi mz2747)  

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 $g$.

DOI: https://doi.org/10.4213/mzm2747

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English version:
Mathematical Notes, 2006, 79:5, 697–706

Bibliographic databases:

UDC: 517.518.32
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

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Golubov B. I., “On approximation by convolutions and bases of shifts of a function”, Anal. Math., 34:1 (2008), 9–28  crossref  mathscinet  zmath  isi  elib  scopus
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