Spectral synthesis in multidimensional Fourier algebras

TL;DR

This paper explores spectral synthesis properties of multidimensional Fourier algebras on locally compact groups, establishing key theorems and connecting synthesis properties across different dimensions.

Contribution

It extends spectral synthesis theory to $A^n(G)$, proving subgroup, injection, inverse projection theorems, and Malliavin's theorem for these algebras.

Findings

Proved subgroup lemma, injection, and inverse projection theorems for spectral and Ditkin sets.

Established parallel synthesis results between $A^n(G)$ and $A^{n+1}(G)$.

Proved Malliavin's theorem in the context of multidimensional Fourier algebras.

Abstract

Let $G$ be a locally compact group and let $A^n(G)$ denote the $n$-dimensional Fourier algebra, introduced by Todorov and Turowska. We investigate spectral synthesis properties of the multidimensional Fourier algebra $A^n(G).$ In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between $A^n(G)$ and $A^{n+1}(G)$ and finally prove Malliavin's theorem.