Coarse-scale representations and smoothed Wigner transforms
arXiv:0804.0259 · doi:10.1016/j.matpur.2009.01.001
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 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 weak-topology approximation result for the Wigner calculus.
58 pages, plain Tex
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Cited by in corpus (11)
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