That's an interesting question, but typically when we deal with periodicity, we need to have some kind of finite period to deal with.
In thinking about this, you might want to check out whether you can do a Fourier series decomposition where the period is infinite (which you should not be able to do) [Also I don't mean Fourier transform, but the decomposition into trig components].
Then you should consider why it's not possible to do this, and if it can be done with some transformation of the function, assumptions, and basis that you project to, then consider what the nature of these changes have to be.