  We analyze energy decay for deep convolutional neural networks employed as
feature extractors, such as Mallat's wavelet scattering transform. For
time-frequency scattering transforms based on Gabor filters, it has been
established that energy decay is exponential, for arbitrary square-integrable
input signals. Our main results allow to prove that this is wrong for wavelet
scattering in arbitrary dimensions. In this setting, the energy decay of the
scattering transform acting on a generic square-integrable signal turns out to
be arbitrarily slow. The fact that this behavior holds for dense subsets of
$L^2(\mathbb{R}^d)$ emphasizes that fast energy decay is generally not a stable
property of signals.
  We complement these findings with positive results allowing to conclude fast
(up to exponential) energy decay for generalized Sobolev spaces that are
tailored to the frequency localization of the underlying filter bank.
  Both negative and positive results highlight that energy decay in scattering
networks critically depends on the interplay of the respective frequency
localizations of the signal on the one hand, and of the employed filters on the
other.
