Mean-Dispersion Principles and the Wigner Transform

Abstract

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