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?