Fourier transform imitations (Q1355213)

Let \(G\) be a finite Abelian group, and \(L(G)\) the space of all complex-valued functions on \(G\). Let \(m\) be the maximal order of an element of \(G\), and \(C_m\) the group of \(m\)th roots of unity. For any \(u\in L(C_m)\) we define \(S_u:L(\widehat G)\to L(G)\) by \(S_uf(\chi)= \sum_{x\in G} u(\chi(-x))f(\chi)\). Observe that, for \(u\equiv 1\), \(S_u\) is the Fourier transform on \(\widehat G\). The paper establishes necessary and sufficient conditions for \(S_u\) to be invertible.