Continuous wavelet transform of Schwartz tempered distributions
Jagdish Narayan Pandey, Santosh Kumar Upadhyay · Cogent Mathematics & Statistics · 2019
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.