Integral transforms and uncertainty products Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often misrepresented.) The Heisenberg uncertainty principle can be arrived at in a couple of ways:


*

*By considering the Cauchy-Schwarz inequality, noting that equality only holds for eigenfunctions of the Fourier transform, knowing a priori what the eigenfunctions of the Fourier transform are and then minimizing the uncertainty product over all eigenfunctions. (The minimizer being the Gaussian of course.)

*Considering commutators of the position and momentum operators.
Many integral transforms that I have come across have uncertainty products associated to them. In the case of the Fourier transform it's not so hard to see since scaling in one domain causes an inverse scaling in the conjugate domain so that the more localized a function is, the more spread out its Fourier transform is. My question is: is this an accident? Or is there something intrinsic about integral transforms on the whole that somehow naturally encodes uncertainty products?
 A: Once i almost had a fight with one of my professors on mathematics because i maintained that this is not accidental. 
The point is to have an understanding of what the fourier transform means for a signal or system and then how this relates to momentum (at least in QM). 
Since fourier transforms (or integral representations in other domains) are used heavily in real-world applications, it is important. 
The intuition is that a signal (or measurement) cannot be localized (arbitrarily and at the same point) in both domains. Roughly because the information for a point at one domain depends on information of multiple points of the other (this is what a linear integral transform does). Mathematically this is expressed as an inequality (upper bound) of the uncertainties of both points and the correlation between the measurements (also called fourier uncertainty principle).
Furthermore one can say that the two measurements cannot be represented exactly on the the same basis (or that the measurements do not commute).
