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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
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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
    
The more focused in freq, the more diffuse in time, that is. –  daniel Oct 25 '11 at 9:56
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Assuming you are looking for a "real-world" application of the mininum-uncertainty property of the Gaussian, I might have one answer:

In quantum mechanics, Gaussians are used to create minimum-uncertainty wavefunctions which are solutions of the Schrödinger equation. The minimum-uncertainty solutions are useful in constructing what are known as coherent states.

See here:

http://en.wikipedia.org/wiki/Coherent_states

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I do not pretend to completely understand the physics but clearly you are right, "minimal uncertainty" (in the sense of my question) is associated with the coherent states described in the article, so this is right on point. Thanks! –  daniel Jun 22 '12 at 23:13
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