Coarse-scale representations and smoothed Wigner transforms

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 GelfandShilov 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 Schrödinger), 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 weaktopology approximation result for the Wigner calculus.