UCLA

\begin{abstract}We show that, with respect to an orthonormal wavelet $\psi(.)\in \L^{2}(\RR),$ any $f(.)\in\L^{2}(\RR)$ is, on the one hand, the sum of its ``layers of details'' over all time-shifts, and on the other hand, the sum of its layers of details over all scales. The latter is well known and is a consequence of a wandering subspace decomposition of $\L^{2}(\RR)$ which, in turn, resulted from a wavelet Multiresolution Analysis (MRA). The former has not been discussed before. We show that it is a consequence of a decomposition of $\L^{2}(\RR)$ in terms of reducing subspaces of the dilation-by-2 shift operator. \end{abstract}