Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral synthesis with the complexity parameter

S. Deodhar, A. Iosevich

Abstract

We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $α$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $κ$, then the classical synthesis threshold improves from $\frac{2d}α$ to $\frac{2(d-2κ)}{α-2κ}$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness

Spectral synthesis with the complexity parameter

Abstract

We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite -dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent , then the classical synthesis threshold improves from to . We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness

Paper Structure

This paper contains 9 sections, 3 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral synthesis in ${\mathbb R}^d$
  3. Sharpness of results
  4. Spectral synthesis on Riemannian manifolds
  5. Outline of the paper
  6. Proof of the main results
  7. Proof of Theorem \ref{['thm:FR_synthesis']}
  8. Proof of Theorem \ref{['thm:manifold-FR-synthesis']}
  9. Concluding remarks

Key Result

Proposition 1.1

Suppose that $\widehat{{(f\mu)}_{R^{-1}}} \equiv \widehat{(f\mu)*\psi_{R^{-1}}}$ is $L^1$-concentrated in $X \subset {\mathbb R}^d$ in the sense that for some $\eta \in (0,1)$, where $X_R=X \cap B_{100R}$. Then where $E_f^{R^{-1}}$ is the $R^{-1}$-neighborhood of the support of $f\mu$. It follows that If we assume that there exist $c, C_X$ universal constants, and $s_f, \alpha_X \in (0,d)$, suc

Figures (2)

  • Figure 1: A schematic illustration of what the Fourier Ratio measures. Diffuse Fourier mass (left) corresponds to a small Fourier-ratio decay exponent $\kappa$, while strong concentration of Fourier mass (right) corresponds to larger $\kappa$ and stronger synthesis conclusions.
  • Figure 2: The synthesis threshold as a function of the Fourier-ratio decay exponent $\kappa$. When $\kappa=0$, one recovers the classical exponent $\frac{2d}{\alpha}$. As $\kappa$ increases, the allowable range of $p$ enlarges, and the threshold diverges as $\kappa \to \frac{\alpha}{2}$.

Theorems & Definitions (9)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4: Localized spectral Fourier ratio
  • Definition 1.5: Neighborhood growth condition
  • Theorem 1.6: Spectral synthesis with the manifold Fourier-ratio parameter
  • Remark 1.7
  • proof
  • proof