In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as ${\mathbf{L}}^q({\mathbb{R}}^n)$ 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 $R\in \text{SO}(n)$.

中文翻译：

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关于非窗口散射变换的泛化

在本文中，我们推广了有限深度小波散射变换，我们将其表述为 ${\mathbf{L}}^q({\mathbb{R}}^n)$ 连续小波变换（或二元小波变换）和收缩非线性的级联范数。然后，我们为这些运算符提供规范，证明这些运算符是明确定义的，并且在特定情况下对 $C^2$ 微分同态的作用是 Lipschitz 连续的。最后，我们扩展了我们的结果，以构建一个对旋转作用不变的算子 $R\in \text{SO}(n)$ 和一个对 的旋转作用等变的算子 $R\in \text{SO}(n)$ 。