the number of non-zero elements in the frequency sequence . Then, This inequality is sharp, with equality achieved when x or X is a Dirac mass, or more generally when x is a nonzero multiple of a Dirac comb supported on a subgroup of the integers modulo N (in which case X is also a Dirac comb supported on a complementary subgroup, and vice versa). More generally, if T and W are subsets of the integers modulo N, let denote the time-limiting operator and band-limiting operators, respectively. Then where the norm is the operator norm of operators on the Hilbert space of functions on the integers modulo N. This inequality has implications for signal reconstruction. When N is a prime number, a stronger inequality holds: Discovered by Terence Tao, this inequality is also sharp. Benedicks's theorem Amrein–Berthier and Benedicks's theorem intuitively says that the set of points where is non-zero and the set of points where is non-zero cannot both be small. Specifically, it is impossible for a function in and its Fourier transform to both be supported on sets of finite Lebesgue measure. A more quantitative version is One expects that the factor may be replaced by , which is only known if either or is convex. Hardy's uncertainty principle The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in is such that and ( an integer), then, if , while if , then there is a polynomial of degree such that This was later improved as follows: if is such that then where is a polynomial of degree and is a real positive definite matrix. This result was stated in Beurling's complete works without proof and proved in Hörmander (the case ) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in ref. A full description of the case as well as the following extension to Schwartz class distributions appears in ref. History Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics. In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad