In the context of harmonic analysis, a branch of mathematics, the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds, ( ∫ ∫ − − ∞ ∞ ∞ ∞ x 2 | f ( x ) | 2 d x ) ( ∫ ∫ − − ∞ ∞ ∞ ∞ ξ ξ 2 | f ^ ^ ( ξ ξ ) | 2 d ξ ξ ) ≥ ≥ ‖ ‖ f ‖ ‖ 2 4 16 π π 2 .