Uncertainty principles

1 Basic Fourier transforms

John D Cook elegantly explains the Fourier uncertainty principle.

2 Signal processing/spectral uncertainties

🏗 see wikipedia for now.

3 Entropic uncertainty

Lazy wikipedia link:

$$g(y)\approx {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp (-2\pi ixy)f(x)dx,f(x)\approx {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp (2\pi ixy)g(y)dy,$$

where the “≈” indicates convergence in $L_2$, and normalized so that (by Plancherel’s theorem),

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|f(x){|}^{2}dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g(y){|}^{2}dy=1.$$

He showed that for any such functions the sum of the Shannon entropies is non-negative,

$$H(|f{|}^2)+H(|g{|}^2)\equiv -{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|f(x){|}^2\log |f(x){|}^{2}dx-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g(y){|}^2\log |g(y){|}^{2}dy\ge 0.$$

$$H(|f{|}^2)+H(|g{|}^2)\ge \log \frac{e}{2},$$

was conjectured by Hirschman and Everett, proven in 1975 by Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The equality holds in the case of Gaussian distributions.

cf the complementarity uncertainty of (Coles, Kaniewski, and Wehner 2014).

The equivalence of entropic uncertainty with wave-particle duality

4 References