Spectral Synthesis on Continuous Images

TL;DR

This paper explores the spectral synthesis property in the Fourier algebra of locally compact Abelian groups, demonstrating that it holds on continuous images of varieties with this property, using the concept of localisability.

Contribution

It introduces the use of localisability of ideals to characterize spectral synthesis on continuous images of varieties, advancing the theoretical understanding in harmonic analysis.

Findings

Spectral synthesis holds on continuous images of varieties with spectral synthesis.

Localisability of ideals characterizes synthesisability in the Fourier algebra.

The work connects localisability with spectral synthesis in a new way.

Abstract

Recently we introduced the concept of localisability of ideals in the Fourier algebra of locally compact Abelian groups. It turns out that localisability can be used to characterise synthesisability of varieties. Based on this we show that spectral synthesis holds on continuous images of varieties which have spectral synthesis.