Uncertainty principle, removing infinity in the Fourier Transform

[ade]

I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold.

Is this a sensible argument?

Thank you.

 

[bhobba]

I presume you mean by the infinity assumption something like it falls away fast enough at infinity for the FT to exist? The distribution approach to FT's means that does not need to hold:
https://www.math.ucdavis.edu/~hunter/book/ch11.pdf

In QM one assumes, when one wants to be rigorous, you are working in a so called Schwartz Space:
https://www.math.ucdavis.edu/~hunter/m218a_09/ch5A.pdf

The uncertainty principle holds for all members of such a space.

Thanks
Bill