Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abelian group G satisfying certain conditions. It has been proved that for every bounded operator M on E commuting with translations there existsh M ∈ L ∞ (G E˜ ) such thatM f ˜ = h Mf ˜ ∀ f ∈ C c ( G ) , whereG E˜is a suitable subset of the group of the continuous morphisms from G into ℂ* andG ˜is a generalized Fourier transform of g defined onG E˜ .