整数采样与Mellin留数的可辨识性

中文摘要 AI 辅助

我证明了所有Mellin留数可由整数样本精确确定，这些样本的误差小于每个代数阶，对于一类显式函数，该类函数由一个具有完整逆幂展开的慢分量和可数多个具有容许包络的载波$\exp(2\pi i\sigma 2^k a^x)$组成。一个单一的例外零测度参数集$a>1$适用于整个函数类。关键估计是长度为$N^{1/2}$的窗口上多项式加权采样Gram矩阵的最终一致下界，具有$O(N^{1/4})$个活跃模式。我从参数平均振荡积分和与乘积Haar测度的有限迹矩比较中推导出该下界，保留了二进频率之间的精确共振。求和Taylor估计然后识别慢渐近展开的每个系数，Mellin延拓论证将这些系数与实际留数等同。我还给出了从现有高阶相关估计的替代推导、一个无限模式示例以及整数参数处的精确反例。该结果为量子引力中Mellin平均启发的模型类提供了插值不变性陈述。

英文摘要

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.

发表机构

• School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院）

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