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(2πiσ2k ax) 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 N1/2, with O(N1/4) 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.
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BibTeX
@misc{he2026integer,
title = {{Integer Sampling and the Identifiability of Mellin Residues}},
author = {Yifan He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38234},
eprint = {2609.38234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38234},
}
RIS
TY - UNPB
TI - Integer Sampling and the Identifiability of Mellin Residues
AU - Yifan He
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.38234
UR - https://arxiv.org/abs/2609.38234
AB - 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(2πiσ2^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.
ER -