Continuous wavelet transform of schwartz distributions

Abstract

In this paper we extend the continuous wavelet transform to Schwartz distributions and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. But the uniqueness theorem for our wavelet inversion formula is valid for the space $\cal D '_F$ obtained by filtering (deleting) (i) all non-zero constant distributions from the space $\cal D '$, (ii) all non-zero constants that appear with a distribution as a union, as for example, for $ {x^2}/(1+x^2) = 1-1/(1+x^2)$, 1 is deleted and $-1/(1+x^2)$ is retained. The kernel of our wavelet transform is an element of $\cal D $ which when integrated along the real line vanishes, but none of its moments of order $m\ge 1$ along the real line is zero. The set of such kernels will be denoted by $D_m$.