Uncertainty Principles in the Euclidean space
Professor Emeritus, University of Orleans, France.
We will start from the classical Heisenberg Inequality of quantum mechanics, which quantifies the uncertainty when measuring simultaneously the position and the momentum of a particle. Heisenberg Inequality may be seen as a property of the Fourier transform, which is also particularly relevant in Signal Processing: a signal cannot be highly localized both in time and in frequency. This non mathematical statement is known as the "Uncertainty Principle". We will then see other mathematical statements, which correspond to different formulations of the Uncertainty Principle, starting from Hardy Uncertainty Principle, which deals with signals that decrease like Gaussian functions, and going to Tao's Uncertainty Principle for finite groups. This last one may be used for sparse signals, in connection with the new theory of "compressed sensing".