# Uncertainty principles

May 14, 2017 — May 14, 2017

## 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

\[ g(y)\approx \int _{{-\infty }}^{\infty }\exp(-2\pi ixy)f(x)\,dx,\qquad f(x)\approx \int _{{-\infty }}^{\infty }\exp(2\pi ixy)g(y)\,dy~, \]

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

\[ \int _{{-\infty }}^{\infty }|f(x)|^{2}\,dx=\int _{{-\infty }}^{\infty }|g(y)|^{2}\,dy=1~. \]

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

\[ H(|f|^{2})+H(|g|^{2})\equiv -\int _{{-\infty }}^{\infty }|f(x)|^{2}\log |f(x)|^{2}\,dx-\int _{{-\infty }}^{\infty }|g(y)|^{2}\log |g(y)|^{2}\,dy\geq 0. \]

\[ H(|f|^{2})+H(|g|^{2})\geq \log {\frac e2}~, \]

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

## 4 References

*Annals of Mathematics*.

*Physical Review E*.

*American Journal of Mathematics*.

*Physical Review Letters*.

*Journal of Functional Analysis*.

*Introduction to Fourier Analysis and Wavelets*. Brooks/Cole Series in Advanced Mathematics.