Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra

Abstract

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 Rouch\'{e}-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.