# Uncertainty principles

May 14, 2017 — May 14, 2017

functional analysis
Hilbert space
probability

## 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

Beckner. 1975. Annals of Mathematics.
Hasegawa, and Van Vu. 2019. Physical Review E.
Hirschman. 1957. American Journal of Mathematics.
Massar, and Spindel. 2008. Physical Review Letters.
Özaydin, and Przebinda. 2004. Journal of Functional Analysis.
Pinsky. 2002. Introduction to Fourier Analysis and Wavelets. Brooks/Cole Series in Advanced Mathematics.
Riegler, and Bölcskei. 2018.