[论文解读] Spectral synthesis on Riemannian manifolds

A. Iosevich, A. Mayeli+1 more|arXiv (Cornell University)|Mar 22, 2026

Numerical methods in inverse problems被引用 0

一句话总结

该论文建立了在紧致 Riemannian 流形上对薄子集支持度量的定量谱合成，呈现环面与球面在几何驱动下的差异，以及稳定性、近似性与不确定性结果。

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.

研究动机与目标

• Motivate and study spectral synthesis for measures supported on thin subsets of compact Riemannian manifolds.
• Establish rigidity and quantitative stability under non-concentration conditions.
• Reveal geometry-dependent behavior of spectral synthesis, with torus vs. sphere contrasts.
• Develop consequences such as approximation by short spectral sums and uncertainty principles.

提出的方法

• Use Laplace–Beltrami spectral projections E_?eras onto eigenfunctions of the Laplacian on a compact manifold M.
• Introduce P_R = () representation via spectral multipliers and finite propagation speed for the wave equation to control support.
• Prove Theorem 1.1 (rigidity) showing that thinly supported measures with finite rac{p} mass vanish for certain p.
• Prove Theorem 1.3 (quantitative stability) giving explicit bounds for <P_R u, chi> in terms of the rac{} norm of u.
• Derive spectral polynomial approximation results (Theorem 1.9) and band-limited converse (Theorem 1.11).
• Establish scale-dependent Fourier ratio bounds (Theorem 1.13) and global upper bounds (Theorem 1.18) relating spectral concentration to uncertainty.

实验结果

研究问题

• RQ1 Under what non-concentration conditions does a measure supported on a thin subset of a compact Riemannian manifold admit unique reconstruction from its spectral data?
• RQ2 How does the geometry of the underlying manifold (torus vs. sphere) affect the rigidity and stability of spectral synthesis?
• RQ3 Can thinly supported signals be stably reconstructed or approximated by low-frequency or short spectral sums?
• RQ4 What are the implications for uncertainty principles and compressed sensing on manifolds?

主要发现

• Synthesis holds on the torus under broad assumptions, yielding rigidity thresholds for spectral mass accumulation.
• On the sphere, synthesis exhibits maximal rigidity, with no finite p-threshold ensuring uniqueness for non-hypersurface supports.
• A quantitative stability bound is established: low-frequency reconstructions P_R u are small if the spectral rac{} mass is small, with explicit rate in R.
• Thin spatial support implies uniform decay estimates for the action of spectral multipliers on u, providing stability across scales.
• Spectral polynomial approximation is guaranteed when the Fourier ratio FR(f) is small, with explicit dependency on the number of eigenfunctions and band-limitation (Theorem 1.9, 1.11).
• An uncertainty principle is formulated: a scale-dependent FR_R(f) is bounded below by geometric quantities, linking eigenfunction growth A() and the measure of the support.