Uncertainty principles

May 14, 2017 — May 14, 2017

functional analysis
Hilbert space

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:

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

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. Inequalities in Fourier Analysis.” Annals of Mathematics.
Hasegawa, and Van Vu. 2019. Uncertainty Relations in Stochastic Processes: An Information Inequality Approach.” Physical Review E.
Hirschman. 1957. A Note on Entropy.” American Journal of Mathematics.
Massar, and Spindel. 2008. Uncertainty Relation for the Discrete Fourier Transform.” Physical Review Letters.
Özaydin, and Przebinda. 2004. An Entropy-Based Uncertainty Principle for a Locally Compact Abelian Group.” Journal of Functional Analysis.
Pinsky. 2002. Introduction to Fourier Analysis and Wavelets. Brooks/Cole Series in Advanced Mathematics.
Riegler, and Bölcskei. 2018. Uncertainty Relations and Sparse Signal Recovery.”