LAUSR

Abstract Let G be a locally compact abelian group and μ be a compactly supported discrete measure on G. We analyse the range of the operator C μ : C ( G ) ⟶ C ( G ) defined by C μ ( f ) ( x ) = ( f ⋆ μ ) ( x ) = ∫ G f ( x − y ) d μ ( y ) . It is shown that this operator is onto when G is a compactly generated locally compact abelian group and μ satisfies certain compatibility conditions. Furthermore, if G is a compactly generated torsion free locally compact abelian group then the convolution operator is always onto for every non zero compactly supported discrete measure μ. For a g ∈ C ( G ) , we construct a function f ∈ C ( G ) such that f ⋆ μ = g .