The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for f and ƒ̂ to both be "very rapidly decreasing". Specifically, if f in L 2 ( R ) is such that | f ( x ) | ≤ ≤ C ( 1 + | x | ) N e − − a π π x 2 and | f ^ ^ ( ξ ξ ) | ≤ ≤ C ( 1 + | ξ ξ | ) N e − − b π π ξ ξ 2 ( C > 0 , N an integer), then, if ab > 1, f = 0, while if ab = 1, then there is a polynomial P of degree ≤ N such that f ( x ) = P ( x ) e − − a π π x 2 .