# Heisenberg Uncertainty Principle

The uncertainty principle (UP) comes up in engineering and physics, but it is a mathematical idea. An old text describes it as "reciprocal spreading." If $f$ is a well-behaved function, the UP might be expressed as $W(f)W(\hat{f}) \geq k$, where $k$ is some constant. If $g$ is a Gaussian, we get equality, i.e., $W(g)W(\hat{g}) = k$.

My question is this. At least in Fourier analysis, the Gaussian is sort of a minimum in the above sense. Are there any real-world problems for which this is a solution? Even in EE I don't think "optimality" of the Gaussian with respect to the UP is ever used.

Thanks for any thoughts.

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A concise statement of the UP, with equality in the case of a Gaussian as an exercise, is in Nievergelt, Wavelets Made Easy, p.236, in case my notation obscures the question. – daniel Oct 24 '11 at 17:02
Hm, maybe I'm missing something but en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle looks like it says the UP is something else. – user12014 Oct 25 '11 at 3:40
Probably the simplest expression of it is given by Linus Pauling in General Chemistry , p.83. dt*dv>=k. I took liberties with the formulation of the idea and it is context-dependent. The expressions used in Nievergelt involve weighted functions of f and its FT. There is a survey in J Fourier Analysis, Nov 3, 1997. – daniel Oct 25 '11 at 9:38
The Wiki article accords with Nievergelt. In words, the more diffuse a function is in the frequency domain, the more focused in the frequency domain and v.v. Perhaps I should have used W(Ff). Hope this clariifies. – daniel Oct 25 '11 at 9:50
I'm not sure I understand the question, but is en.wikipedia.org/wiki/Gabor_transform relevant? – endolith Oct 27 '11 at 0:28