Periodic Integral Transforms and C*-algebras

We construct canonical integral transforms, analogous to the Fourier transform, that have periods six and three. The existence of such transforms is shown to arise naturally from the expectation that the Schwartz space on the real line, viewed as the Heisenberg module of Rieffel and Connes over the rotation C*-algebra, should extend to a module action over the crossed product of the latter by the canonical automorphisms of orders three and six (which does in fact happen and is shown here).