Localization operators of the wavelet transform associated to the Riemann–Liouville operator

2016 ◽  
Vol 27 (04) ◽  
pp. 1650036 ◽  
Author(s):  
C. Baccar ◽  
N. B. Hamadi
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Liouville Operator ◽  
Continuous Wavelet ◽  
Localization Operators ◽  
Von Neumann ◽  
Neumann Class

We study the continuous wavelet transform [Formula: see text] associated with the Riemann–Liouville operator. Next, we investigate the localization operators for [Formula: see text]; in particular we prove that they are in the Schatten-von Neumann class.

Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator

2019 ◽  
Vol 17 (04) ◽  
pp. 1950022 ◽  
Author(s):  
Hatem Mejjaoli ◽  
Nadia Ben Hamadi ◽  
Slim Omri
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Finite Measure ◽  
Continuous Wavelet ◽  
Uncertainty Principles ◽  
Localization Operators ◽  
Spherical Mean Operator ◽  
Von Neumann ◽  
Time Frequency ◽  
Spherical Mean

We consider the continuous wavelet transform [Formula: see text] associated with the spherical mean operator. We investigate the localization operators for [Formula: see text], in particular, we prove that they are in the Schatten-von Neumann class. Next, we analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given. Finally, we prove many versions of quantitative uncertainty principles for [Formula: see text].

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

Sign in / Sign up

Export Citation Format

Share Document