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Spectral Synthesis on Varieties

László Székelyhidi

TL;DR

This paper generalizes spectral synthesis for varieties from the real line to locally compact Abelian groups by employing localization of Fourier-algebra ideals via abstract derivations on compactly supported measures. It introduces generalized differential operators on the Fourier algebra, and proves a key equivalence: the annihilator of an ideal is synthesizable if and only if the ideal is localizable, linking synthesis to the multiplicity data encoded by roots in $Z(\\widehat{I})$. The work further shows that non-polynomial derivations obstruct localizability, provides discrete-group criteria based on torsion-free rank, and leverages reductions to $G/B$ to move toward a complete characterization of spectral synthesis on LCAs. Overall, the framework connects exponential monomials, differential-operator structure, and localization to establish precise conditions under which spectral synthesis holds, offering a path to a full classification among locally compact Abelian groups.

Abstract

In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense subspace. He studied so-called local ideals in the space of Fourier transforms, and his proof based on the observation that, on the one hand, these local ideals are completely determined by the exponential monomials in the space, and, on the other hand, these local ideals completely determine the space itself. On the other hand, Dimitri Gurevich gave counterexamples for Schwartz's theorem in higher dimension. In this paper we show that the ideas of localisation can be extended to general locally compact Abelian groups using abstract derivations on the Fourier algebra of compactly supported measures. Based on this method we present necessary and sufficient conditions for spectral synthesis for varieties on locally compact Abelian groups. Using localisation, in \cite{MR4789359} we proved that spectral synthesis holds on a locally compact Abelian group $G$ if and only if it holds on $G/B$, where $B$ is the closed subgroup of compact elements. This may lead to a complete characterisation of locally compact Abelian groups having spectral synthesis.

Spectral Synthesis on Varieties

TL;DR

This paper generalizes spectral synthesis for varieties from the real line to locally compact Abelian groups by employing localization of Fourier-algebra ideals via abstract derivations on compactly supported measures. It introduces generalized differential operators on the Fourier algebra, and proves a key equivalence: the annihilator of an ideal is synthesizable if and only if the ideal is localizable, linking synthesis to the multiplicity data encoded by roots in . The work further shows that non-polynomial derivations obstruct localizability, provides discrete-group criteria based on torsion-free rank, and leverages reductions to to move toward a complete characterization of spectral synthesis on LCAs. Overall, the framework connects exponential monomials, differential-operator structure, and localization to establish precise conditions under which spectral synthesis holds, offering a path to a full classification among locally compact Abelian groups.

Abstract

In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense subspace. He studied so-called local ideals in the space of Fourier transforms, and his proof based on the observation that, on the one hand, these local ideals are completely determined by the exponential monomials in the space, and, on the other hand, these local ideals completely determine the space itself. On the other hand, Dimitri Gurevich gave counterexamples for Schwartz's theorem in higher dimension. In this paper we show that the ideas of localisation can be extended to general locally compact Abelian groups using abstract derivations on the Fourier algebra of compactly supported measures. Based on this method we present necessary and sufficient conditions for spectral synthesis for varieties on locally compact Abelian groups. Using localisation, in \cite{MR4789359} we proved that spectral synthesis holds on a locally compact Abelian group if and only if it holds on , where is the closed subgroup of compact elements. This may lead to a complete characterisation of locally compact Abelian groups having spectral synthesis.
Paper Structure (6 sections, 8 theorems, 27 equations)

This paper contains 6 sections, 8 theorems, 27 equations.

Table of Contents

  1. Abstract
  2. Introduction
  3. Differential operators on the Fourier algebra
  4. Localizable ideals
  5. Spectral synthesis and localizability
  6. Statements and Declarations

Key Result

Theorem 1

Let $G$ be a locally compact Abelian group. The continuous function $f:G\to \mathbb{C}$ is an exponential polynomial if and only if it has the form where $a_i$ is a real additive function, $m_j:G\to \mathbb{C}$ is an exponential, and $P_j:\mathbb{C}^k\to \mathbb{C}$ is a polynomial $(i=1,2,\dots,k; j=1,2,\dots,n)$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • proof
  • Corollary 4
  • ...and 1 more