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Spectral synthesis on Riemannian manifolds

A. Iosevich, A. Mayeli, E. Wyman

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.

Spectral synthesis on Riemannian manifolds

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.
Paper Structure (19 sections, 10 theorems, 126 equations, 1 figure)

This paper contains 19 sections, 10 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Uniqueness and stability
  3. Implications for compressed sensing on manifolds
  4. Sharpness and lack thereof
  5. Explicit example on $S^2$
  6. Spectral polynomial approximation
  7. Properties of the Fourier ratio and the uncertainty principle
  8. Proofs of the main results
  9. Proof of Theorem \ref{['thm:stability']}
  10. Proof of Theorem \ref{['main']}
  11. Proof of Proposition \ref{['prop:sphere-lack-sharpness']}
  12. Proof of Theorem \ref{['theorem:manifoldfourierratio']}
  13. Proof of Theorem \ref{['thm:FR-converse-bandlimited']}
  14. Proof of Theorem \ref{['theorem:localFRlowerbound']}
  15. Proof of Theorem \ref{['theorem:manifold-FR-sandwich']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $M$ be a compact Riemannian manifold without boundary, and let $u$ be a Radon measure carried by a $C^1$ submanifold of $M$ of dimension $k$, $1 \le k < d$. If for some $p$ with $2 \le p \le \frac{2d}{k}$, then $u$ is identically $0$.

Figures (1)

  • Figure 1: Conceptual illustration of spectral synthesis on a compact Riemannian manifold $M$. (Left) A thinly supported measure $u$ on a submanifold $S \subset M$ and its spectral projections $E_\lambda u$. (Middle) Spectral mass $\|E_\lambda u\|_{L^2}$ vs. frequency $\lambda$: rapid $\ell^p$ decay (Case B) forces $u=0$ under $p \leq 2d/k$ (Theorem \ref{['main']}). (Right) The Fourier ratio $\mathrm{FR}(f)$ controls approximation by short spectral sums (Theorem \ref{['theorem:manifoldfourierratio']}). Bottom: key concepts and geometry-dependent sharpness.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Quantitative stability for thinly supported signals
  • Remark 1.4: Relation to signal recovery
  • Remark 1.5: Conceptual picture
  • Proposition 1.6: Lack of sharpness on the sphere
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9
  • Definition 1.10: Band-limited functions
  • ...and 19 more