Approximations by convolutions and antiderivatives

A. M. Sedletskii

Matematičeskie zametki, Tome 79 (2006) no. 5, pp. 756-766

Résumé

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 $g$.