Continuous wavelet transform of Schwartz tempered distributions

• Main Authors: J.N. Pandey, S.K. Upadhyay
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2019-01-01
• Series: Cogent Mathematics & Statistics
• Subjects: primary: 46f12; secondary: 46f05.
• Online Access: http://dx.doi.org/10.1080/25742558.2019.1623647

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 $$S'({\mathbb R})$$. But uniqueness theorem for the present wavelet inversion formula is valid for the space $${S'_F}({\mathbb R})$$ obtained by filtering (deleting) (i) all non-zero constant distributions from the space $$S'({\mathbb R})$$, (ii) all non-zero constants that appear with a distribution as a union. As an example, in considering the distribution $${{{x^2}} \over {1 + {x^2}}} = 1 - {1 \over {1 + {x^2}}}$$ we would omit 1 and retain only $$ - {1 \over {1 + {x^2}}}$$. The wavelet kernel under consideration for determining the wavelet transform are those wavelets whose all the moments are non-zero. As an example, $$(1 + kx - 2{x^2}){e^{ - {x^2}}}$$ 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.