On reconstruction from discrete local moving averages on locally compact abelian groups

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 Cμ:C(G)⟶C(G) defined by Cμ(f)(x)=(f⋆μ)(x)=∫Gf(x−y)dμ(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.