Continuous wavelet transform of schwartz tempered distributions in S'(ℝ n )

Abstract

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. © 2019 by the authors.