CONTINUOUS WAVELET TRANSFORM OF SCHWARTZ DISTRIBUTIONS

摘要

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 D-F' obtained by filtering (deleting) (i) all non-zero constant distributions from the space 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 D which when integrated along the real line vanishes, but none of its moments of order m >= 1 along the real line is zero. The set of such kernels will be denoted by D-m.