an:03890108
Zbl 0558.43002
Rezola, M. L.
A theorem of density for translation invariant subspaces of \(L^ p(G)\)
EN
Boll. Unione Mat. Ital., VI. Ser., A 4, 43-47 (1985).
00150071
1985
j
43A15
dense subspaces; locally compact Abelian Hausdorff group; translation invariant subspaces
Given a locally compact Abelian Hausdorff group G with Haar measure, and denoting by \(L_ p(G)\) the corresponding Banach spaces, the author proves three theorems assuring the density of translation invariant subspaces S of \(L_ p(G)\) for \(1\leq p<\infty\), under some additional assumptions (among them, invariance of S under multiplication with suitable functions). We state the last theorem: If S is a self-adjoint translation invariant subspace of \(L_ p(G)\) and there exists \(\phi \in L_{\infty}(G)\) which is not periodic and such that \(\phi\) \(S\subseteq S\), then S is dense in \(L_ p(G)\).
G.Crombez