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).

scholarly journals Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 235 ◽  
Author(s):  
Jagdish Pandey ◽  
Jay Maurya ◽  
Santosh Upadhyay ◽  
Hari Srivastava
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Weak Topology ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Wavelet Inversion ◽  
Constant Distribution

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

The continuous wavelet transform in n-dimensions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650037 ◽  
Author(s):  
J. N. Pandey ◽  
N. K. Jha ◽  
O. P. Singh
Keyword(s):  
Image Processing ◽  
Wavelet Transform ◽  
Real Number ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Spherically Symmetric ◽  
Continuous Wavelet ◽  
Wavelet Inversion ◽  
Translation Parameter

Daubechies obtained the [Formula: see text]-dimensional inversion formula for the continuous wavelet transform of spherically symmetric wavelets in [Formula: see text] with convergence interpreted in the [Formula: see text]-norm. From the wavelet [Formula: see text], Daubechies generated a doubly indexed family of wavelets [Formula: see text] by restricting the dilation parameter [Formula: see text] to be a real number greater than zero and the translation parameter [Formula: see text] belonging to [Formula: see text]. We show that [Formula: see text] can be chosen to be in [Formula: see text] with none of the components [Formula: see text] vanishing. Further, we prove that if [Formula: see text] and [Formula: see text] are continuous in [Formula: see text], then the convergence besides being in [Formula: see text] is also pointwise in [Formula: see text]. We advance our theory further to the case when [Formula: see text] and [Formula: see text] both belong to [Formula: see text] then convergence of the wavelet inversion formula is pointwise at all points of continuity of [Formula: see text]. This result significantly enhances the applicability of the wavelet inversion formula to the image processing.

scholarly journals Continuous Wavelet Transform of Schwartz Tempered Distributions in $S'(\mathbb R^n)$

2019 ◽  
Author(s):  
Jagdish Narayan Pandey ◽  
Jay Singh Maurya ◽  
Santosh Kumar Upadhyay ◽  
Hari Mohan Srivastava
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Weak Topology ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Wavelet Inversion ◽  
Constant Distribution

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

BESOV NORMS IN TERMS OF THE CONTINUOUS WAVELET TRANSFORM. APPLICATION TO STRUCTURE FUNCTIONS

1996 ◽  
Vol 06 (05) ◽  
pp. 649-664 ◽  
Author(s):  
VALÉRIE PERRIER ◽  
CLAUDE BASDEVANT
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Besov Spaces ◽  
Inversion Formula ◽  
Structure Functions ◽  
Continuous Wavelet ◽  
Wavelet Coefficients ◽  
Small Scales ◽  
Besov Norms

The continuous wavelet transform is extended to Lp spaces and an inversion formula is demonstrated. From this the Besov spaces can be characterized by the behavior at small scales of the wavelet coefficients. These results apply to the measurement of structure functions.

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.

INVERSION OF THE RADON TRANSFORM ASSOCIATED WITH THE CLASSICAL DOMAIN OF TYPE ONE

2005 ◽  
Vol 16 (08) ◽  
pp. 875-887 ◽  
Author(s):  
JIANXUN HE ◽  
HEPING LIU
Keyword(s):  
Wavelet Transform ◽  
Differential Operator ◽  
Lie Group ◽  
Radon Transform ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Weak Sense ◽  
Continuous Wavelet ◽  
Nilpotent Lie Group ◽  
Classical Domain

Let D(Ω,Φ) be the unbounded realization of the classical domain [Formula: see text] of type one. In general, its Šilov boundary [Formula: see text] is a nilpotent Lie group of step two. In this article we define the Radon transform on [Formula: see text], and obtain an inversion formula [Formula: see text] in terms of a determinantal differential operator. Moreover, we characterize a subspace of [Formula: see text] on which the Radon transform is a bijection. By use of the suitable continuous wavelet transform we establish a new inversion formula of the Radon transform in weak sense without the assumption of differentiability.

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