Abstract
We consider the continuous wavelet transform mathcal{S}_{h}^{W} associated with the Weinstein operator. We introduce the notion of localization operators for mathcal {S}_{h}^{W}. In particular, we prove the boundedness and compactness of localization operators associated with the continuous wavelet transform. Next, we analyze the concentration of mathcal{S}_{h}^{W} on sets of finite measure. In particular, Benedicks-type and Donoho-Stark’s uncertainty principles are given. Finally, we prove many versions of Heisenberg-type uncertainty principles for mathcal{S}_{h}^{W}.
Highlights
IntroductionWe consider the Weinstein operator ( called the Laplace-Bessel differential operator (see [ ])) defined on Rd– × ( , ∞) by d ∂ β + ∂ β := i= ∂x i + xd ∂xd
In this paper, we consider the Weinstein operator ( called the Laplace-Bessel differential operator) defined on Rd– × (, ∞) by d ∂ β + ∂ β := i= ∂x i + xd ∂xd =x + Lβ,xd, β>, where x is the Laplace operator on Rd, and Lβ,xd the Bessel operator on (, ∞) given by d β + d Lβ,xd := dx d + xd, dxd β>– . ( . )The Weinstein operator β has several applications in pure and applied mathematics especially in fluid mechanics.The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem [, ]
As the harmonic analysis associated with the Weinstein operator has known remarkable development, it is a natural question whether there exist an equivalent of the theory of localization operators and new uncertainty principles for the continuous wavelet transform relating to this harmonic analysis
Summary
We consider the Weinstein operator ( called the Laplace-Bessel differential operator (see [ ])) defined on Rd– × ( , ∞) by d ∂ β + ∂ β := i= ∂x i + xd ∂xd. In particular the authors have introduced and studied the generalized Fourier transform associated with the Weinstein operator. As the harmonic analysis associated with the Weinstein operator has known remarkable development, it is a natural question whether there exist an equivalent of the theory of localization operators and new uncertainty principles for the continuous wavelet transform relating to this harmonic analysis. We want to study the localization operators for the continuous Weinstein wavelet transform. In Section , we introduce and study the two-localization operators associated with the Weinstein continuous wavelet transform. In Section , we study the quantitative analysis of the continuous Weinstein wavelet transform and time-frequency concentration. The Weinstein continuous wavelet transform ShW on Rd+ is defined for regular functions f on Rd+ by. The transformation ShW is a bounded linear operator from L β (Rd+) into the space of continuous bounded functions on d+.
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