  In this paper, we generalize finite depth wavelet scattering transforms,
which we formulate as $\Lb^q(\mathbb{R}^n)$ norms of a cascade of continuous
wavelet transforms (or dyadic wavelet transforms) and contractive
nonlinearities. We then provide norms for these operators, prove that these
operators are well-defined, and are Lipschitz continuous to the action of $C^2$
diffeomorphisms in specific cases. Lastly, we extend our results to formulate
an operator invariant to the action of rotations $R \in \text{SO}(n)$ and an
operator that is equivariant to the action of rotations of $R \in
\text{SO}(n)$.
