For a phenomenon $\pmb{f}$ that is a function of $\mathit{n}$ factors, defined on a finite abelian group $\mathcal{G}$, we derive its population statistics solely from its Fourier transform $\hat{\pmb{f}}$. Our main result is an $\mathit{m-Coefficient/Index Annihilation Theorem}$: the $\mathit{m}$th moment of $\pmb{f}$ becomes a series of terms, each with precisely $\mathit{m}$ Fourier coefficients -- and surprisingly, the coefficient $\mathit{indices}$ in each term sum to zero under group addition. This condition acts like a filter, limiting which terms appear in the Fourier domain, and can reveal deeper relationships between the variables driving $\pmb{f}$. These techniques can also be used as an analytical/design tool, or as a feasibility constraint in search algorithms. For functions defined on $\mathbb{Z}_2^n$, we show how the skew, kurtosis, etc. of a binomial distribution can be derived from the Fourier domain. Several other examples are presented.

翻译：对于定义在有限阿贝尔群 $\mathcal{G}$ 上、依赖于 $\mathit{n}$ 个因子的现象 $\pmb{f}$，我们仅从其傅里叶变换 $\hat{\pmb{f}}$ 推导出其总体统计量。我们的主要结果是 $\mathit{m}$ 系数/索引湮灭定理：$\pmb{f}$ 的 $\mathit{m}$ 阶矩可表示为一系列项的和，其中每一项恰好包含 $\mathit{m}$ 个傅里叶系数——且令人惊讶的是，每一项中的系数$\mathit{索引}$在群加法下之和为零。这一条件如同一个滤波器，限制了傅里叶域中出现的项，并可揭示驱动 $\pmb{f}$ 的变量之间的深层关系。这些技术还可作为分析/设计工具，或作为搜索算法中的可行性约束。对于定义在 $\mathbb{Z}_2^n$ 上的函数，我们展示了如何从傅里叶域导出二项分布的偏度、峰度等统计量。文中还给出了其他若干示例。