Two geometric interpretations of the multidimensional Hardy uncertainty principle

Abstract

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. • We state a multidimensional version of Hardy's uncertainty principle. • The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid. • We express this result in terms of the projections of the Hardy ellipsoid on the x - and p -spaces.