Spectral Synthesis in the Multidimensional Fourier Algebra and the Varopolous Algebra for Compact Groups

TL;DR

This paper explores the structure of multidimensional Fourier and Varopolous algebras on compact groups, establishing an embedding and extending synthesis results in harmonic analysis.

Contribution

It introduces a multidimensional Varopolous algebra and demonstrates its embedding of the Fourier algebra, extending synthesis results in the context of compact groups.

Findings

Established the embedding of multidimensional Fourier algebra into the Varopolous algebra.

Proved a parallel synthesis result generalizing previous work.

Unified the understanding of spectral synthesis in these algebras.

Abstract

Let $G$ be a compact group and let $A^n(G)$ denote the multidimensional Fourier algebra introduced by Todorov and Turowska. In this note, we first define the multidimensional version of the Varopolous algebra and show that the multidimensional Fourier algebra can be embedded into the multidimensional Varopolous algebra. Using this embedding, we also prove a result on parallel synthesis, subsuming the earlier results of Varopolous, Spronk-Turowska and Parthsarathy-Prakash.