An uncertainty inequality

Recently, the authors derived an uncertainty relationship between noise and spatial resolution in linear computational imaging systems that is similar to the Heisenberg uncertainty principle for conjugate variables in quantum mechanics. Specifically, for linear shift-invariant systems with a fixed incident photon density, the product of noise and spatial resolution has a positive absolute lower limit which can be evaluated with the aid of the inequality ‖ g ‖ p + 1 a ( p + 1 ) ‖ | x | a g ‖ 1 n p ‖ g ‖ 1 a ( p + 1 ) + n p ≥ C ( n , p , a ) , p > 0 ; a > 0 , where g ∈ L 1 ( R n ) ∩ L 1 ( R n , | x | a ) ∩ L p + 1 ( R n ) . This inequality, which is derived in this note, also has applications in data smoothing and in variations of Heisenberg’s inequality.