Wavelet multipliers and signals

Abstract

The Schatten-von Neumann property of a pseudo-differential operator is established by showing that the pseudo-differential operator is a multiplier defined by means of an admissible wavelet associated to a unitary representation of the additive group R" on the C*-algebra of all bounded linear operators from L 2 (R") into L 2 (K"). A bounded linear operator on L 2 (IR) arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier studied in this paper.

Wavelet Transforms and Localization Operators

“…In another direction, guided by the Landau-Pollak-Slepian operator in signal analysis, a theory of wavelet multipliers has been initiated in the paper [41] by He and Wong, developed in the paper [22] by Du and Wong, and detailed in the book [104] by Wong. Wavelet multipliers are localization operators on the additive group IR n defined in terms of coherent states parametrized by points in IR n .…”

𝐿^{𝑝} boundedness of localization operators associated to left regular representations

“…That the Landau-Pollak-Slepian operator in signal analysis (studied in the fundamental papers [12,13] by Landau and Pollak, [14,15] by Slepian, and [16] by Slepian and Pollak) is in fact a wavelet multiplier is a result proved in [10] by He and Wong, and in [20] by Wong. As in the case of localization operators, it is proved in [10] by He and Wong, and in [20] …”

“…In another direction, guided by the Landau-Pollak-Slepian operator in signal analysis, a theory of wavelet multipliers has been introduced and studied in [6] by Du and Wong,[10] by He and Wong and [20] …”

“…Then it is proved in [10] by He and Wong, and in [20] by Wong that the bounded linear operators P σ,ϕ :…”

Wavelet multipliers on 𝐿^{𝑝}(ℝⁿ)

“…See, for instance, the works [6], [8], [9] and [12]. In [11], the L p -boundedness of localization operators associated to left regular representations is studied for 1 ≤ p ≤ ∞.…”