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.

Wavelet transforms associated with the Kontorovich–Lebedev transform

2017 ◽  
Vol 15 (02) ◽  
pp. 1750011 ◽  
Author(s):  
Akhilesh Prasad ◽  
U. K. Mandal
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Wavelet Transforms ◽  
Discrete Version ◽  
Continuous Wavelet ◽  
Reconstruction Formula ◽  
Basic Properties

The main objective of this paper is to study continuous wavelet transform (CWT) using the convolution theory of Kontorovich–Lebedev transform (KL-transform) and discuss some of its basic properties. Plancherel’s as well as Parseval’s relation and Reconstruction formula for CWT are obtained and some examples are also given. The discrete version of the wavelet transform associated with KL-transform is also given and reconstruction formula is derived.

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 on local fields

2015 ◽  
Vol 34 (2) ◽  
pp. 113-121 ◽  
Author(s):  
Ashish Pathak
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Local Fields ◽  
Continuous Wavelet ◽  
Mother Wavelet ◽  
Basic Properties

The main objective of this paper is to define the mother wavelet on local fields and study the continuous wavelet transform (CWT) and some of their basic properties. its inversion formula, the Parseval relation and associated convolution are also studied.

scholarly journals On alternative wavelet reconstruction formula: a case study of approximate wavelets

2014 ◽  
Vol 1 (2) ◽  
pp. 140124 ◽  
Author(s):  
Elena A. Lebedeva ◽  
Eugene B. Postnikov
Keyword(s):  
Wavelet Transform ◽  
Wide Class ◽  
Continuous Wavelet Transform ◽  
Morlet Wavelet ◽  
Continuous Wavelet ◽  
Reconstruction Formula ◽  
Physical Processes ◽  
Admissibility Condition ◽  
Oscillatory Dynamics

The application of the continuous wavelet transform to the study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies owing to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard reduced Morlet wavelet, which is an important example of such analysing functions, 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).

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