Uncertainty Principle, annihilating pairs and Fourier restriction

TL;DR

This paper strengthens uncertainty principle estimates for functions on locally compact abelian groups by incorporating restriction theorems, providing explicit bounds for the constants involved, with applications in signal processing and control theory.

Contribution

It introduces improved uncertainty estimates when subsets satisfy restriction theorems, extending previous results to finite groups and offering explicit constant bounds.

Findings

Enhanced uncertainty estimates under restriction conditions.

Extension of results to finite groups.

Explicit bounds for the constants in inequalities.

Abstract

Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S \subset G$ and $\Sigma \subset \widehat{G}$, there exists a constant $C(S, \Sigma)$ such that for any $f \in L^2(G)$, the following inequality holds: \[\|f\|_{L^2(G)} \leq C(S, \Sigma) \bigl( \|f\|_{L^2(G \setminus S)} + \|\widehat{f}\|_{L^2(\widehat{G} \setminus \Sigma)} \bigr),\] where $\widehat{f}$ denotes the Fourier transform of $f$. This variant of the uncertainty principle is particularly useful in applications such as signal processing and control theory.The purpose of this paper is to show that such estimates can be strengthened when $S$ or $\Sigma$ satisfies a restriction theorem and to provide an estimate for the constant $C(S, \Sigma)$. This result serves as a quantitative counterpart to a recent finding by the first and last author. In the setting of finite groups, the results also extend those of Matolcsi-Sz\"ucs and Donoho-Stark.