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
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