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The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

A. Iosevich, Z. Li, E. Palsson, A. Yavicoli

TL;DR

This work develops a continuous analogue of the Fourier ratio for compactly supported measures, coupling scale-regularized Fourier transforms with geometric covering data to form a fractal uncertainty principle. It shows that small Fourier ratio implies low-complexity approximations by low-degree trigonometric polynomials and provides constructive random Fourier sampling schemes for such approximations, with degree bounds tied to covering numbers and Minkowski dimensions. The theory differentiates deterministic curvature-driven measures (e.g., circle arc, convex surfaces) from random fractal measures (e.g., Laba–Wang Cantor), yielding sharp dichotomies in restriction-type phenomena and approximation capabilities. Applications span exact signal recovery under sparse spectral data, convex-geometry analysis, and a unified view of discrete vs. continuous Fourier ratio phenomena, enriching both harmonic analysis and geometric measure theory.

Abstract

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure \(μ\) on \(\mathbb{R}^d\) and \(f\in L^2(μ)\), the Fourier ratio compares \(L^1\) and \(L^2\) norms of a regularized Fourier transform at scale \(R\). We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in \(L^1\), \(L^2\), and \(L^\infty\). In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.

The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

TL;DR

This work develops a continuous analogue of the Fourier ratio for compactly supported measures, coupling scale-regularized Fourier transforms with geometric covering data to form a fractal uncertainty principle. It shows that small Fourier ratio implies low-complexity approximations by low-degree trigonometric polynomials and provides constructive random Fourier sampling schemes for such approximations, with degree bounds tied to covering numbers and Minkowski dimensions. The theory differentiates deterministic curvature-driven measures (e.g., circle arc, convex surfaces) from random fractal measures (e.g., Laba–Wang Cantor), yielding sharp dichotomies in restriction-type phenomena and approximation capabilities. Applications span exact signal recovery under sparse spectral data, convex-geometry analysis, and a unified view of discrete vs. continuous Fourier ratio phenomena, enriching both harmonic analysis and geometric measure theory.

Abstract

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure on and \(f\in L^2(μ)\), the Fourier ratio compares and norms of a regularized Fourier transform at scale . We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in , , and . In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.

Paper Structure

This paper contains 41 sections, 21 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. The Core Problem and Main Definition
  3. Summary of Main Contributions
  4. Fractal Uncertainty Principle and Exact Recovery
  5. Approximation by Low-Degree Trigonometric Polynomials
  6. Restriction Theory and the Deterministic--Random Dichotomy
  7. Convex Geometry and the Complexity of Surface Measures
  8. Connections and Broader Context
  9. Structure of the Paper
  10. Notation and Conventions
  11. Previous results
  12. Bounding the Fourier ratio from below.
  13. The main results of A2025
  14. Lower and upper bounds on the Fourier ratio
  15. An upper bound on the Fourier ratio and a fractal uncertainty principle
  16. ...and 26 more sections

Key Result

Theorem 2.1

Let $(\varphi_j)_{j=1}^n$ be an orthonormal system in $L^2(\mathbb{Z}_N)$ with $\|\varphi_j\|_{L^\infty} \leq K$ for $1 \leq j \leq n$. There exists a constant $\gamma_0 \in (0,1)$ and a subset $I \subset \{1, \dots, n\}$ with $|I| \ge \gamma_0 n$ such that for every $a = (a_i) \in \mathbb{C}^n$, where $C_T > 0$ is a universal constant.

Theorems & Definitions (31)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 21 more