Mean-dispersion principles and the Wigner transform

Abstract

Given a function $f\in L^2(\mathbb{R})$, we consider means and variances associated to $f$ and its Fourier transform $\hat{f}$, 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.

31 pages

References in corpus (3)

• Uncertainty principles for orthonormal sequences
• Quantum States and Hardy's Formulation of the Uncertainty Principle : a Symplectic Approach
• Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms