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.

Distribution space, Distributions, Fourier transform, Function spaces and their duals, Generalized functions, Schwartz testing function space, Tempered distributions, Wavelet transform of generalized functions
dx.doi.org/10.3390/sym11020235
Symmetry
School of Mathematics and Statistics

Pandey, J. N, Maurya, J.S. (Jay Singh), Upadhyay, S.K. (Santosh Kumar), & Srivastava, H.M. (Hari Mohan). (2019). Continuous wavelet transform of schwartz tempered distributions in S'(ℝ n ). Symmetry, 11(2). doi:10.3390/sym11020235