Meanwhile, I have run down to the library to get hold of Heisenberg's original article Zs. Phys. 33 (1925) 879-893 in English translation (thank you, anonymous) from the volume Sources of Quantum Mechanics, edited by van der Waerden and published by North-Holland in 1967.
As anonymous has noted, Heisenberg was indeed thinking in terms of Fourier transforms, not necessarily commutative, and his original approach to quantum mechanics takes the measurement geometry philosophy seriously. To quote: "instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur." Heisenberg's relations look very categorical in nature. His first example is the category of emission frequencies for an electron. In two short pages he argues that the phases are of just as much significance in the quantum case as in the classical. In terms of the real part of expressions $U(n, n - a)exp(i \omega (n, n - a)t)$ he says: "Only the origin of the time scale and hence a phase factor common to all the $U$ is arbitrary and accordingly devoid of physical significance, but the phases of the individual $U$ enter in an essential manner into the [induced scattering moment]."