Linear Perturbations of the Wigner Transform and the Weyl Quantization

Abstract

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.The group of invertible, real-valued 2d × 2d matrices is denoted byand we denote the transpose of an inverse matrix byLet J denote the canonical symplectic matrix in R 2d , namelyObserve that, for z = (z 1 , z 2 ) ∈ R 2d , we have Jz = J (z 1 , z 2 ) = (z 2 , −z 1 ) , J −1 z = J −1 (z 1 , z 2 ) = (−z 2 , z 1 ) = −Jz, and J 2 = −I 2d×2d . Function spacesRecall that C 0 (R d ) denotes the class of continuous functions on R d vanishing at infinity. Modulation spaces. Fix a non-zero window g ∈ S(R d ) and 1 ≤ p, q ≤ ∞., (with obvious modifications for p = ∞ or q = ∞). If p = q, we write M p instead of M p, p .

Dispersion, spreading and sparsity of Gabor wave packets for metaplectic and Schrödinger operators

“…We remark that t → S t coincides with the Hamiltonian flow for the free particle in phase space; precisely, the classical equations of motion with Hamiltonian H(x, ξ) = |ξ| 2 and initial datum (x 0 , ξ 0 ) ∈ R 2d are solved by (x(t), ξ(t)) = S t (x 0 , ξ 0 ). Hence (1) shows that the time evolution of wave packets under U(t) approximately follows the classical flow, in according with the correspondence principle of quantum mechanics. Nevertheless, a distinctive feature of wave propagation dynamics is the unavoidable effect of diffraction.…”

“…It may therefore appear quite unsatisfactory that there is no trace of such issues in quasi-diagonalization estimates as (1). The purpose of this paper is exactly to prove refined estimates for the Gabor matrix of U(t) where sparsity, spreading and dispersive phenomena are fully represented.…”

Wigner transform and quasicrystals

“…This situation is more involved but still some results can be obtained. Furthermore we shall consider a generalization of the Wigner transform, called matrix-Wigner transform, see [1], which, using a composition with linear maps, will yield a unifying framework connecting our results to those of [7], [8]. We need some preliminaries.…”

“…As mentioned above, following [1], we introduce next a Matrix-Wigner transform which is a natural generalization of the Wigner transform W (µ, ν) = F 2 (T s (µ ⊗ ν)) where the change of coordinates T s have been replaced by a general bijective linear map T . This sesquilinear transform turns out to be a quite comprehensive tool including most of the basic time-frequency representations, we refer to [1] for details and properties. Definition 11.…”

On Inverses of the Dirac Comb

“…The expression f , ϕ ≡ f (ϕ) is the application of f to some test function ϕ. It naturally extends the usual inner product on Lebesgue-square integrable functions to the space of tempered distributions [23,[63][64][65][66][67]. The expression f ≡ f , 1 denotes the integral of f in the tempered distributions sense [4,9] and the integral symbol f (t) dt is only used for ordinary Lebesgue integrable functions f (t).…”