In Fourier theory, the pair composed of a variable and its Fourier transform is called conjugate variables, and one crucial property between the two is the uncertainty relation. This relation tells us that e.g. if one variable/function has a bounded (or compact?) support, its Fourier transform cannot have a bounded support as well. Now in the context of Quantum Mechanics, as an example, the physical variables are represented by linear operators, and there we have the famed Heisenberg relations for special pairs of quantities, such as position and momentum, or two different spin components. The question is, do we have the same bounded and unboundedness consequence of Fourier transforms when we talk about linear operators such as the ones in QM? (so Hermitian ones).
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