On spectral synthesis in ${\mathbb{Z}}_N^d$

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 $p\le \frac{2n}{d}$, 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.