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 as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The equality holds in the case of Gaussian distributions.