散射变换的通用容许性
Universal admissibility for scattering transforms
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中文总结 AI 辅助
该研究解决散射变换的容许性问题，证明特定Parseval滤波器组可诱导保范散射变换，建立残差范数衰减界，并构造反例说明无法成立通用指数衰减率。
中文摘要 AI 辅助
我们解决了散射变换的容许性问题：对ℝᵈ上的每个低通滤波器在原点处非零的Parseval滤波器组，无需任何解析性、频率局部化或几何覆盖假设，即可诱导出保范散射变换。此外，对任意Sobolev输入f∈Hˢ(ℝᵈ)，我们建立了深度为N的残差范数的通用衰减界为𝒪(N⁻ᵐⁱⁿ{ˢ,¹}/ᵈ)。最后，我们构造了一个带限Schwartz函数的滤波器组和一个带限Schwartz输入，其低通乘子在原点附近等于1，但传播能量呈亚指数衰减，这表明仅在这些假设下无法成立通用指数衰减率。
英文摘要
We resolve the admissibility problem for scattering transforms: every Parseval filter bank on $\mathbb{R}^d$ whose low-pass filter is nonvanishing at the origin induces a norm-preserving scattering transform, without requiring any analyticity, frequency-localization, or geometric covering assumptions. Furthermore, for any Sobolev input $f\in H^s(\mathbb{R}^d)$, we establish a universal decay bound of $\mathcal{O}(N^{-\min\{s,1\}/d})$ on the depth-$N$ residual norm. Finally, we construct a filter bank of band-limited Schwartz functions and a band-limited Schwartz input for which the low-pass multiplier equals one near the origin but the propagated energy decays subexponentially. This demonstrates that no universal exponential rate can hold under these assumptions alone.
