Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5773550 | Applied and Computational Harmonic Analysis | 2017 | 11 Pages |
â¢We state a multidimensional version of Hardy's uncertainty principle.â¢The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid.â¢We express this result in terms of the projections of the Hardy ellipsoid on the x- and p-spaces.
Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry.