Spectral synthesis on Riemannian manifolds

TL;DR

This paper investigates spectral synthesis for measures on compact Riemannian manifolds, revealing how geometric differences influence synthesis and establishing stability bounds, with applications to approximation and uncertainty principles.

Contribution

It provides the first comprehensive analysis of spectral synthesis on Riemannian manifolds, highlighting geometric dependencies and offering explicit stability bounds.

Findings

Spectral synthesis holds on tori under broad conditions.

On the sphere, spectral synthesis can fail sharply.

The results unify spectral synthesis with geometric and stability considerations.

Abstract

We study spectral synthesis for measures supported on thin subsets of compact Riemannian manifolds. We prove that under natural non-concentration conditions, such measures admit quantitative spectral synthesis, with explicit stability bounds. We show that this phenomenon depends strongly on the underlying geometry. On the torus, synthesis holds under broad assumptions, while on the sphere we establish rigidity results demonstrating that synthesis can fail in a sharp sense. As consequences, we obtain quantitative approximation results and uncertainty principles for functions with thin spectral support. These results provide a unified framework connecting spectral synthesis, geometric structure, and stability on compact manifolds.