Continuous wavelet transform of Schwartz tempered distributions

Abstract

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 . But uniqueness theorem for the present wavelet inversion formula is valid for the space obtained by filtering (deleting) (i) all non-zero constant distributions from the space , (ii) all non-zero constants that appear with a distribution as a union. As an example, in considering the distribution we would omit 1 and retain only . The wavelet kernel under consideration for determining the wavelet transform are those wavelets whose all the moments are non-zero. As an example, is such a wavelet. 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.