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Continuous wavelet transform of Schwartz tempered distributions.

Authors :
Pandey, J.N.
Upadhyay, S.K.
Liu, Lishan
Source :
Cogent Mathematics & Statistics. Jan2019, Vol. 6 Issue 1, p1-15. 15p.
Publication Year :
2019

Abstract

The continuous wavelet transform of Schwartz tempered distributions is investigated and derive the corresponding wavelet inversion formula (valid modulo a constant-tempered distribution) interpreting convergence in S ′ (R). But uniqueness theorem for the present wavelet inversion formula is valid for the space S F ′ (R) obtained by filtering (deleting) (i) all non-zero constant distributions from the space S ′ (R) , (ii) all non-zero constants that appear with a distribution as a union. As an example, in considering the distribution x 2 1 + x 2 = 1 − 1 1 + x 2 we would omit 1 and retain only − 1 1 + x 2 . The wavelet kernel under consideration for determining the wavelet transform are those wavelets whose all the moments are non-zero. As an example, (1 + k x − 2 x 2) e − x 2 is such a wavelet. k is an arbitrary constant. There exist many other classes of such wavelets. In our analysis, we do not use a wavelet kernel having any of its moments zero. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
25742558
Volume :
6
Issue :
1
Database :
Academic Search Index
Journal :
Cogent Mathematics & Statistics
Publication Type :
Academic Journal
Accession number :
141152021
Full Text :
https://doi.org/10.1080/25742558.2019.1623647