Huh. Handwave-y Claim: The fourier transform is a logical connective. Proof: (One version of) the fourier transform relates a sequence of coefficients c_n to Σ_n c_n e^{pi i x n}. Substituting u = e^{pi i x}, this is just an ordinary power series Σ_n c_n u^n ...
You want theta functions now. Start with the Jacobian theta functions (warning: this is going to skyrocket the amount of abstraction you're going to be able to deal with), take three b3nadr<l.