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A theorem of density for translation invariant subspaces of \(L^ p(G)\). (English) Zbl 0558.43002
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 \(L_ p(G)\).
Reviewer: G.Crombez
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.