Statistics of a multi-factor function from its Fourier transform

Abstract

For a phenomenon $\mathbf{f}$ that is a function of $n$ factors, defined on a finite abelian group $G$, we derive its population statistics solely from its Fourier transform $\hat{\mathbf{f}}$. Our main result is an $m$-Coefficient/Index Annihilation Theorem: the $m$th moment of $\mathbf{f}$ becomes a series of terms, each with precisely $m$ Fourier coefficients --- and surprisingly, the coefficient 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 $\mathbf{f}$. 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.

Submitted to the Journal of Fourier Analysis and Applications. 42 pages, 6 figures