On Sets of Large Fourier Transform Under Changes in Domain

TL;DR

This paper studies how large Fourier coefficients of a function over one domain relate to those over a different domain, showing they remain large and localized, which aids in coefficient recovery and re-proving known results.

Contribution

It establishes that large Fourier coefficients are preserved and localized under domain changes, enabling coefficient recovery and providing a new proof for a known result.

Findings

Large Fourier coefficients are preserved under domain change.

Large coefficients are contained in small intervals around scaled indices.

The results facilitate coefficient recovery and reprove existing theorems.

Abstract

A function $f:\mathbb{Z}_n \to \mathbb{C}$ can be represented as a linear combination $f(x)=\sum_{\alpha \in \mathbb{Z}_n}\widehat{f}(\alpha) \chi_{\alpha,n}(x)$ where $\widehat{f}$ is the (discrete) Fourier transform of $f$. Clearly, the basis $\{\chi_{\alpha,n}(x):=\exp(2\pi i \alpha x/n)\}$ depends on the value $n$. We show that if $f$ has "large" Fourier coefficients, then the function $\widetilde{f}:\mathbb{Z}_m \to \mathbb{C}$, given by \[ \widetilde{f}(x) = \begin{cases} f(x) & \text{when } 0\leq x < \min(n, m), 0 & \text{otherwise}, \end{cases} \] also has "large" coefficients. Moreover, they are all contained in a "small" interval around $\lfloor \frac{m}{n}\alpha \rceil$ for each $\alpha \in \mathbb{Z}_n$ such that $\widehat{f}(\alpha)$ is large. One can use this result to recover the large Fourier coefficients of a function $f$ by redefining it on a convenient domain. One can also use this result to reprove a result by Morillo and R{\`a}fols: \emph{single-bit} functions, defined over any domain, have a small set of large coefficients.