A wavelet “time-shift-detail” decomposition

We show that, with respect to an orthonormal wavelet ψ(·)∈L2(R) any f(·)∈L2(R) 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 L2(R) 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 L2(R) in terms of reducing subspaces of the dilation-by-2 shift operator.