Wavelet filtering with the Mellin transform (Q1392813)

The author suggests some technical improvements to wavelet theory. First, given a basic wavelet \(\psi(t)\), the corresponding wavelet family is defined as \(\psi_{\sigma, \tau} (t)= \psi (\sigma t-\tau)\); \(\sigma, \tau \in \mathbb{R}\), \(\sigma\neq 0\), which is equivalent to the usual \(\psi_{s,\tau} (t)= | s|^{-1/2} \psi ((t- \tau)/s)\), but more simple. The (continuous) wavelet transform then is \[ W_{\sigma, \tau}(\chi) =\langle \psi_{\sigma, \tau}, \chi \rangle \int_\mathbb{R} \chi (t) \psi(\sigma t-\tau)^*dt \] and the parameter \(\sigma= 1/s\) is interpreted as frequency scale. Next, certain representations of the convolution operator are obtained using Fourier and Mellin transforms. In the author's interpretation, any convolution operator in the time domain can be represented as a multiplication operator in the wavelet (time scale) domain. It is argued that such representations are useful for many applications.