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The Principal Ideal Theorem in Spectral Synthesis

László Székelyhidi

TL;DR

This work addresses the Principal Ideal Theorem within spectral synthesis on locally compact abelian groups by employing a localization framework for ideals in the Fourier algebra. It shows that localizability of an ideal is equivalent to the synthesizability of its annihilator, linking algebraic structure to density of exponential monomials solving convolution equations $\mu*f=0$. The main achievement is a transparent, general proof that every locally compact abelian group is a PIT-group, reducing to $\mathbb{R}^n$ and using quotient and product structures to extend the result to all lca groups. The findings deepen the connection between algebraic localization, polynomial derivations on the Fourier algebra, and spectral synthesis, providing robust tools for analyzing exponential monomials and the density of solutions to convolution equations.

Abstract

In an earlier paper we solved a long-standing problem which goes back to Laurent Schwartz's work on mean-periodic functions. Namely, we completely characterised those locally compact Abelian groups having spectral synthesis. The method is based on the localisation concept. In this paper we show that localisation can be used to prove another basic result in spectral synthesis: the principal ideal theorem.

The Principal Ideal Theorem in Spectral Synthesis

TL;DR

This work addresses the Principal Ideal Theorem within spectral synthesis on locally compact abelian groups by employing a localization framework for ideals in the Fourier algebra. It shows that localizability of an ideal is equivalent to the synthesizability of its annihilator, linking algebraic structure to density of exponential monomials solving convolution equations . The main achievement is a transparent, general proof that every locally compact abelian group is a PIT-group, reducing to and using quotient and product structures to extend the result to all lca groups. The findings deepen the connection between algebraic localization, polynomial derivations on the Fourier algebra, and spectral synthesis, providing robust tools for analyzing exponential monomials and the density of solutions to convolution equations.

Abstract

In an earlier paper we solved a long-standing problem which goes back to Laurent Schwartz's work on mean-periodic functions. Namely, we completely characterised those locally compact Abelian groups having spectral synthesis. The method is based on the localisation concept. In this paper we show that localisation can be used to prove another basic result in spectral synthesis: the principal ideal theorem.
Paper Structure (5 sections, 8 theorems, 16 equations)

This paper contains 5 sections, 8 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Differential operators on the Fourier algebra
  3. Localization
  4. Roots of principal ideals on $\mathbb{R}^n$
  5. Statements and Declarations

Key Result

Theorem 1

For any nonzero linear partial differential operator $P(D)$ on $\mathbb{R}^n$, the exponential monomials solving that solve the equation $P(D)f = 0$ linearly span a dense subspace of the solution space.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • proof
  • ...and 1 more