Algebraic structures for $\bigoplus \sum \sb{n\geq 1}L\sp{2}(Z/n)$ compatible with the finite Fourier transform

Let ${Z / n}$ denote the integers $\bmod\,n$ and let ${\mathcal{F}_n}$ denote the finite Fourier transform on ${L^2}({Z / n})$. We let $\oplus \Sigma {{\mathcal{F}_n}} = F$ operate on $\oplus \Sigma {L^2}({Z / n})$ and show that $\oplus \Sigma {L^2}({Z / n})$ can be given a graded algebra structure (with no zero divisors) such that $\mathcal{F}(fg) = \mathcal{F}(f)\mathcal{F}(g)$. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on $\oplus \Sigma {L^2}({Z / n})$ satisfying the above condition.