AI 中文总结
本文提出膨胀协变Hankel铅笔方法，利用多尺度几何样本直接生成Hankel结构，消除对数混叠，并通过轮廓绕数与Rouché型界实现稀疏Mellin谱的鲁棒恢复，数值实验验证了其提高可辨识性与分辨率的优势。
AI 中文摘要
我们研究从几何样本中恢复稀疏Mellin谱的问题。研究表明，膨胀协变性直接生成Hankel结构，而非仅在简化为经典指数和模型之后才生成。对于复指数，不可公度的采样尺度消除了单尺度下持续存在的对数混叠。从最小Hankel铅笔可获得第二条恢复通道：轮廓绕数在区域上计数谱节点，而Rouché型界在噪声下认证孤立节点和未分辨簇。相应的轮廓裕度在采样比趋近1时具有显式的退化速率。数值实验表明，辅助尺度同时提高了可辨识性和分辨率，且认证的轮廓计数在逐点精确恢复失效的范围内仍可保持可靠。
英文摘要
We study sparse Mellin spectral recovery from geometric samples. Dilation covariance is shown to generate the Hankel structure directly, rather than only after reduction to a classical exponential-sum model. For complex exponents, incommensurate sampling scales remove the logarithmic aliasing that persists at a single scale. A second recovery channel is obtained from the minimal Hankel pencil: contour winding counts spectral nodes regionally, while Rouché-type bounds certify isolated nodes and unresolved clusters under noise. The associated contour margin has an explicit degeneration rate as the sampling ratio approaches one. Numerical experiments show that the auxiliary scale improves both identifiability and resolution, and that certified contour counts can remain reliable beyond the regime of accurate pointwise recovery.
