Coarse-scale representations and smoothed Wigner transforms

TL;DR

This paper develops explicit formulations for coarse-scale smoothed Wigner transforms and operators, enabling phase-space reformulations of linear and nonlinear wave equations with controlled resolutions, and shows their approximation by differential operators in the semiclassical limit.

Contribution

It introduces a framework for coarse-scale smoothed Wigner calculus, deriving explicit formulas for smoothed operators and reformulating wave equations as phase-space equations with adjustable resolutions.

Findings

Explicit formulas for smoothed operators of infinite order.

Reformulation of wave equations as coarse-scale phase-space equations.

Approximation of smoothed Wigner calculus by differential operators in semiclassical regime.

Abstract

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting ``smoothed operators'' are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand-Shilov spaces, is necessary. Similarly we treat the ``smoothed Wigner calculus''. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g. Hartree and cubic non-linear Schr\"odinger), as coarse-scale phase-space equations (e.g. smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.