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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
MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc.