Discretization, sampling, and the Fourier ratio

TL;DR

This paper establishes fundamental sampling bounds for smooth signals, linking spectral compressibility induced by smoothness to recoverability from incomplete samples, and provides explicit bounds for functions on domains and the sphere.

Contribution

It introduces the Fourier ratio as a spectral compressibility measure and derives deterministic bounds connecting smoothness to sampling requirements for signal recovery.

Findings

Random samples suffice for accurate recovery of smooth signals

Sample complexity scales polylogarithmically with bandwidth

Smoothness enforces Fourier domain compressibility

Abstract

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework.