Quaternionic continuous wavelet transform on a quaternionic Hilbert space

2017 ◽  
Vol 112 (4) ◽  
pp. 1049-1057 ◽  
Author(s):  
M. Fashandi
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Continuous Wavelet ◽  
Quaternionic Hilbert Space

THE IMAGE SPACE OF ONE TYPE OF CONTINUOUS WAVELET TRANSFORM AND ITS PROPERTY

2012 ◽  
Vol 10 (03) ◽  
pp. 1250022 ◽  
Author(s):  
CAIXIA DENG ◽  
ZUOXIAN FU ◽  
SHUAI LI
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Fundamental Theorem ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Image Space ◽  
Continuous Wavelet ◽  
Admissible Wavelet ◽  
Linear Transform

In this paper, we show that the space of continuous wavelet transform is a reproducing kernel Hilbert space based on the fundamental theorem of linear transform. An admissible wavelet is got by convolution computation which is made into continuous wavelet transform. By the theory of reproducing kernel we can discuss correlative properties of image space of wavelet transform, which provide theoretic frame for us to study image space of the general wavelet transform.

ON THE CONTINUOUS WAVELET TRANSFORM ON HOMOGENEOUS SPACES

2012 ◽  
Vol 10 (04) ◽  
pp. 1250038 ◽  
Author(s):  
F. ESMAEELZADEH ◽  
R. A. KAMYABI GOL ◽  
R. RAISI TOUSI
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Reproducing Kernel ◽  
Homogeneous Spaces ◽  
Reproducing Kernel Hilbert Space ◽  
Compact Subgroup ◽  
Continuous Wavelet ◽  
Square Integrable Representation ◽  
Necessary And Sufficient

Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space [Formula: see text]. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto.

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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