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

TL;DR

This paper introduces a continuous Fourier ratio for compactly supported measures, establishing a fractal uncertainty principle that links measure support complexity with approximation and recovery capabilities.

Contribution

It develops a new Fourier ratio concept, proves a sharp fractal uncertainty principle, and explores implications for signal recovery and measure approximation.

Findings

Sharp bounds for the Fourier ratio in terms of covering numbers

Small Fourier ratio implies efficient low-degree polynomial approximation

Restriction estimates differentiate curved measures from fractal measures

Abstract

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure \(\mu\) on \(\mathbb{R}^d\) and \(f\in L^2(\mu)\), 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.