AI 中文总结
该研究探讨现代调和分析中多尺度表示的分辨率与鲁棒性平衡问题，基于波包系统确定分辨率 - 鲁棒性权衡，建立几何刚性现象，还找出索伯列夫阈值，为欧几里得散射变换提供变形稳定性理论及首个稳定性估计。
AI 中文摘要
现代调和分析中的一个核心挑战是量化精细分辨的多尺度表示的逼近能力与其对坐标非线性变化的鲁棒性之间的平衡，这一问题在信号处理和偏微分方程中自然出现。受Mallat关于小波散射变换的开创性结果的启发，我们确定了基于一般波包系统构建的散射型非线性多尺度表示的尖锐分辨率 - 鲁棒性权衡，表明在小微分同胚下的稳定性由底层频率分解的几何结构决定。特别是对于具有比小波更精细横向分辨率的波包系统，包括曲波和剪切波，我们建立了一种几何刚性现象：任意小的、光滑的、紧支集变形可以使高频质量跨越相邻通道，在第一散射层就导致不稳定性。我们通过确定变形稳定性的尖锐索伯列夫阈值来补充这一障碍：低于临界正则性，不存在Mallat型估计成立，而在临界正则性及以上，通过匹配的换位子界恢复稳定性，该界允许变形通过频率通道传播。这些结果共同为欧几里得散射变换提供了一个系统的变形稳定性理论，并给出了超越经典小波设置的散射架构固有的第一个稳定性估计。
英文摘要
A central challenge in modern harmonic analysis is to quantify the balance between the approximation power of finely resolved multiscale representations and their robustness to nonlinear changes of coordinates, a problem arising naturally in signal processing and partial differential equations. Motivated by Mallat's pioneering results on the wavelet scattering transform, we identify a sharp resolution--robustness trade-off for scattering-type nonlinear multiscale representations built upon general wave-packet systems, showing that stability under small diffeomorphisms is governed by the geometry of the underlying frequency decomposition. In particular, for wave-packet systems with finer transverse resolution than wavelets, including curvelets and shearlets, we establish a geometric rigidity phenomenon: arbitrarily small, smooth, compactly supported deformations can move high-frequency mass across adjacent channels, leading to instability already at the first scattering layer. We complement this obstruction by identifying the sharp Sobolev threshold for deformation stability: below the critical regularity no Mallat-type estimate can hold, while at and above it stability is recovered by means of matched commutator bounds that allow deformations to be propagated through the frequency channels. Together, these results provide a systematic deformation-stability theory for Euclidean scattering transforms and yield the first stability estimates intrinsic to the scattering architecture beyond the classical wavelet setting.
Comments55 pages; 2 figures
