On reconstruction from discrete local moving averages on locally compact abelian groups (Q1754673)

Let \(G\) be a locally compact abelian group and let \(M_{cd}(G)\) be the set of all compactly supported discrete Borel measures \(\mu\) on \(G\). The paper is devoted to studying the range of the convolution operator \[ C_\mu:\;C(G)\to C(G),\;\;(C_\mu f)(x)=(f*\mu)(x)=\int_G f(x-y)d\mu(y), \;\;x\in G. \] Sufficient conditions for the surjectivity of the map \(C_\mu\) are obtained. In particular, if \(G\) is a compactly generated locally compact abelian group having no proper compact subgroups, then for every nonzero measure \(\mu\in M_{cd}(G)\) and every function \(g\in C(G)\) there exists a function \(f\in C(G)\) such that \(f*\mu=g\). This holds if \(G\in\{\mathbb R^n,\mathbb Z^m,\mathbb R^n\times \mathbb Z^m\}\), where \(n,m\in\{0,1,2,\ldots\}\).