Sketch of Proof
The proof of this tight inequality depends on the so-called (q, p)-norm of the Fourier transformation. (Establishing this norm is the most difficult part of the proof.) From this norm we are able to establish a lower bound on the sum of the (differential) Rényi entropies where For simplicity, we consider this inequality only in one dimension; the extension to multiple dimensions is straightforward and can be found in the literature cited.
Read more about this topic: Entropic Uncertainty
Famous quotes containing the words sketch and/or proof:
“the vagabond began
To sketch a face that well might buy the soul of any man.
Then, as he placed another lock upon the shapely head,
With a fearful shriek, he leaped and fell across the
picture—dead.”
—Hugh Antoine D’Arcy (1843–1925)
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)