Wiener Tauberian theorems for vector-valued functions (Q1335496)

Summary: Different versions of Wiener's Tauberian theorem are discussed for the generalized group algebra \(L^1 (G,A)\) (of integrable functions on a locally compact abelian group \(G\) taking values in a commutative semisimple regular Banach algebra \(A)\) using \(A\)-valued Fourier transforms. A weak form of a Wiener's Tauberian property is introduced and it is proved that \(L^1 (G,A)\) is weakly Tauberian if and only if \(A\) is. The vector analogue of Wiener's \(L^2\)-span of translates theorem is examined.