# Uncertainty principles

## Basic Fourier transforms

John D Cook elegantly explains the Fourier uncertainty principle.

## Signal processing/spectral uncertainties

🏗 see wikipedia for now.

## 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.

## References

Beckner, William. 1975. Annals of Mathematics 102 (1): 159–82.
Hasegawa, Yoshihiko, and Tan Van Vu. 2019. Physical Review E 99 (6): 062126.
Hirschman, I. I. 1957. American Journal of Mathematics 79 (1): 152–56.
Massar, Serge, and Philippe Spindel. 2008. Physical Review Letters 100 (19): 190401.
Özaydin, Murad, and Tomasz Przebinda. 2004. Journal of Functional Analysis 215 (1): 241–52.
Pinsky, Mark A. 2002. Introduction to Fourier Analysis and Wavelets. Brooks/Cole Series in Advanced Mathematics. Australia ; Pacific Grove, CA: Brooks/Cole.

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