Mean-dispersion principles and the Wigner transform

Given a function $f\in L^2(\mathbb{R})$, we consider means and variances associated to $f$ and its Fourier transform $\widehat{f}$, and explore their relations with the Wigner transform $W(f)$, obtaining a simple new proof of Shapiro’s mean-dispersion principle. Uncertainty principles for orthonormal sequences in $L^2(\mathbb{R})$ 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.