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Open AccessArticle

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

1
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
2
Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India
3
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
4
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
*
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(2), 235; https://doi.org/10.3390/sym11020235
Received: 11 January 2019 / Revised: 2 February 2019 / Accepted: 12 February 2019 / Published: 15 February 2019
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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. View Full-Text
Keywords: function spaces and their duals; distributions; tempered distributions; Schwartz testing function space; generalized functions; distribution space; wavelet transform of generalized functions; Fourier transform function spaces and their duals; distributions; tempered distributions; Schwartz testing function space; generalized functions; distribution space; wavelet transform of generalized functions; Fourier transform
MDPI and ACS Style

Pandey, J.N.; Maurya, J.S.; Upadhyay, S.K.; Srivastava, H.M. Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n ). Symmetry 2019, 11, 235.

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