Some Results in Spectral Synthesis Over ${\mathbb Z}_N^d$

TL;DR

This paper explores spectral synthesis on finite cyclic groups, extending classical Fourier support results to discrete settings and connecting with signal recovery and combinatorial constructions.

Contribution

It adapts classical spectral synthesis results to functions on finite cyclic groups, integrating Bourgain's $ ext{Lambda}_p$ sets and random methods.

Findings

Extension of spectral support results to ${f Z}_N^d$

Connections established with signal recovery theory

Incorporation of Bourgain's $ ext{Lambda}_p$ sets and random constructions

Abstract

A classical result due to Agranovsky and Narayanan (\cite{AN04}) says that if the support of the Fourier transform of $f: {\mathbb R}^n \to {\mathbb C}$ is carried by a smooth measure on a $d$-dimensional manifold $M$, and $f \in L^p({\mathbb R}^d)$ for $p \leq \frac{2n}{d}$, then $f$ is identically equal to $0$. In this paper, we investigate an analogous problem for functions $f: {\mathbb Z}_N^d \to {\mathbb C}$. Bourgain's celebrated result on $\Lambda_p$ sets (\cite{Bou89}), random constructions (\cite{Bab89}), and connections with the theory of exact signal recovery (\cite{DS89}, \cite{MS73}, \cite{IKLM24}, \cite{IM24}) play an important role.