Continuous wavelet transform involving canonical convolution

2019 ◽  
Vol 13 (06) ◽  
pp. 2050104
Author(s):  
Zamir Ahmad Ansari
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Convolution Operator ◽  
Linear Canonical Transform ◽  
Continuous Wavelet ◽  
Test Function ◽  
Test Function Space

The main objective of this paper is to study the continuous wavelet transform in terms of canonical convolution and its adjoint. A relation between the canonical convolution operator and inverse linear canonical transform is established. The continuity of continuous wavelet transform on test function space is discussed.

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).

Continuous wavelet transform associated with zero-order Mehler–Fock transform and its composition

2018 ◽  
Vol 16 (06) ◽  
pp. 1850050 ◽  
Author(s):  
Akhilesh Prasad ◽  
S. K. Verma
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Convolution Operator ◽  
Zero Order ◽  
Continuous Wavelet ◽  
Reconstruction Formula ◽  
Basic Properties ◽  
Time Invariant

The continuous wavelet transform (CWT) associated with zero-order Mehler–Fock transform (MF-transform) is defined and discussed its some basic properties, Plancherel’s and Parseval’s relations, reconstruction formula for CWT are obtained. Further composition of CWT is investigated and then its Parseval’s and Plancherel’s relations are given. Moreover, time-invariant filter has been defined and proved convolution operator and wavelet transform are represented as time-invariant transform.

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