Generalized Localization Principle for Continuous Wavelet Decompositions

2019 ◽  
Vol 106 (5-6) ◽  
pp. 857-863
Author(s):  
R. R. Ashurov ◽  
Yu. É. Faiziev
Keyword(s):  
Continuous Wavelet ◽  
Localization Principle ◽  
Wavelet Decompositions

scholarly journals Continuous Wavelet Decompositions, Multiresolution, and Contrast Analysis

1993 ◽  
Vol 24 (3) ◽  
pp. 739-755 ◽  
Author(s):  
M. Duval-Destin ◽  
M. A. Muschietti ◽  
B. Torresani
Keyword(s):  
Continuous Wavelet ◽  
Contrast Analysis ◽  
Wavelet Decompositions

CONVERGENCE OF THE CONTINUOUS WAVELET TRANSFORMS ON THE ENTIRE LEBESGUE SET OF Lp-FUNCTIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 675-683 ◽  
Author(s):  
RAVSHAN ASHUROV
Keyword(s):  
Lebesgue Measure ◽  
Wavelet Transforms ◽  
Measure Zero ◽  
The Other ◽  
Continuous Wavelet ◽  
Exceptional Set ◽  
Localization Principle ◽  
Continuous Wavelet Transforms ◽  
Almost Everywhere ◽  
Lebesgue Set

The almost everywhere convergence of wavelets transforms of Lp-functions under minimal conditions on wavelets is well known. But this result does not provide any information about the exceptional set (of Lebesgue measure zero), where convergence does not hold. In this paper, under slightly stronger conditions on wavelets, we prove convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. On the other hand, practically all the wavelets, including Haar and "French hat" wavelets, used frequently in applications, satisfy our conditions. We also prove that the same conditions on wavelets guarantee the Riemann localization principle in L1 for the wavelet transforms.

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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