[gizmo686]

Your can replace the real numbers with the rational numbers and maintain all of the vector structure.

If you wanted something more quantized, you can pick some length unit, d, and replace the real numbers with {... -2d, -d, 0, d, 2d,... }. This forms a structure known as a "ring" with the standard notion of addition, subtraction, and multiplication (but no notion of division. Using this instead of R does lose the vector structure, but is still an example of a slightly more general notion of a "module". Many of the linear algebra results for vector spaces apply to modules as well.

> If it isn't possible, what similar construction on the fuzzy numbers would get you to the obvious endpoint of a "fourier analysis with a low pass filter pre-applied?"

If that is where you want to end up, you could pretty much start there. If you take all real value functions and apply a courier analysis with a low pass filter to each of them, the resulting set still forms a vector space. Although I don't see any particular way of arriving at this vector space by manipulating functions pre Fourier transform.