Mean-Dispersion Principles and the Wigner Transform

Given a function \(f\in L^2(\mathbb {R})\) $f\in L^2(\mathbb{R})$ , we consider means and variances associated to f and its Fourier transform \(\hat{f}\) $\widehat{f}$ , and explore their relations with the Wigner transform W(f), obtaining, as particular cases, a simple new proof of Shapiro’s mean-dispersion principle, as well as a stronger result due to Jaming and Powell. Uncertainty principles for orthonormal sequences in \(L^2(\mathbb {R})\) $L^2(\mathbb{R})$ involving linear partial differential operators with polynomial coefficients and the Wigner distribution, or different Cohen class representations, are obtained, and an extension to the case of Riesz bases is studied.