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). 



Does the uncertainty relation of Fourier transforms also extend to linear operators?
Yes, the Heisenberg uncertainty principle literally is the Fourier uncertainty relation. For simplicity consider the one-dimensional case.


The Hilbert space of a spinless one-dimensional particle is $L^2(\mathbb{R})$ and position is represented by $Q = M\_x$, the operator of multiplication by $x$. Whereas momentum is represented by $P = -i\hbar \frac{d}{dx}$, which is just to say that $P = \hbar F^{-1}M\_xF$ where $F: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is the $L^2$-normalized Fourier transform. Heisenberg's uncertainty principle is then a fact about the trade-off between the variance of a wave function $\psi \in L^2(\mathbb{R})$ and the variance of its Fourier transform, because the position variance of $\psi$ is $$\|M\_x\psi\|\_2^2 - \langle M\_x\psi,\psi\rangle^2 = \int x^2|\psi(x)|^2 - \left(\int x|\psi(x)|^2\right)^2$$ and its momentum variance is $$\left\|-i\hbar\frac{d}{dx} \psi\right\|\_2^2 - \left\langle -i\hbar\frac{d}{dx}\psi,\psi\right\rangle = \hbar^2\int x^2|\hat{\psi}(x)|^2 - \hbar^2\left(\int x|\hat{\psi}(x)|^2\right)^2.$$


You can also see from this that, yes, a particle cannot simultaneously have a bounded range of possible position values and a bounded range of possible momentum values --- that is just saying that $\psi$ and $\hat{\psi}$ cannot both be supported on bounded intervals.


