Integer Sampling and the Identifiability of Mellin Residues

Abstract

I prove that all Mellin residues are determined exactly by integer samples known up to errors smaller than every algebraic order, for an explicit class of functions consisting of a slow component with a complete inverse-power expansion and countably many carriers $\exp (2\pi i\sigma 2^k{a}^x)$ with admissible envelopes. A single exceptional null set of parameters $a>1$ works for the entire function class. The key estimate is an eventual uniform lower bound for a polynomially weighted sampling Gram matrix on windows of length $N^{1/2}$, with $O(N^{1/4})$ active modes. I derive this bound from parameter-averaged oscillatory integrals and a finite trace-moment comparison with product Haar measure, retaining the exact resonances among dyadic frequencies. Summed Taylor estimates then identify every coefficient of the slow asymptotic expansion, and a Mellin continuation argument identifies these coefficients with the actual residues. I also give an alternative derivation from existing higher-correlation estimates, an infinite-mode example, and exact counterexamples at integer parameters. The result supplies an interpolation-invariance statement for a model class motivated by Mellin averaging in quantum gravity.