A Mathematical Theory of Near-Field Super-Resolution

2026-09-25

AI Summary

This study addresses the geometric challenge of super-resolution arising from wavefront curvature in finite-aperture near-field sensing by proposing a deterministic recovery theory for sparse measures based on a quadratic phase aperture criterion. Methodologically, it replaces the classical minimum separation condition with a support-uniform quadratic phase criterion and constructs a finite harmonic Bessel–Vandermonde lifting structure. This framework integrates total variation minimization, non-asymptotic quadratic sum bounds, and norm-bounded Hermite dual certificate techniques. The proposed theory achieves exact reconstruction uniformly over the entire support. Furthermore, it naturally degenerates into Fourier-type angular super-resolution in the far-field limit, thereby demonstrating exact recovery capability for arbitrary non-zero amplitudes.

Abstract

Finite-aperture near-field sensing leads to a super-resolution geometry fundamentally different from the translation-invariant Fourier setting. In the Fresnel regime, wavefront curvature introduces a range-dependent quadratic aperture phase, so distinguishability is governed by incomplete quadratic exponential sums of the form \( \sum_{n=0}^{N_r-1} a_n e^{i(ω_1 n+ω_2 n^2)} \), rather than by angular separation alone. We develop a deterministic recovery theory for sparse measures with ranges on a finite grid and continuous angles, and introduce a support-uniform quadratic-phase aperture criterion replacing classical minimum separation. Under this criterion, total-variation minimization exactly recovers every sparse measure in the admissible support class, uniformly over all nonzero complex amplitudes. The proof develops nonasymptotic support-uniform bounds for finite quadratic sums and combines them with a gauged Hermite dual certificate controlling interpolation, local curvature, and off-support leakage. We further construct a finite-harmonic Bessel-Vandermonde lift with explicit truncation error. In the far-field limit, the quadratic phase disappears and the theory reduces to Fourier-type angular super-resolution.

Problem

• near-field super-resolution
• finite-aperture sensing
• quadratic phase
• sparse recovery
• Fresnel regime

Innovation

• Near-field super-resolution
• Quadratic exponential sums
• Total-variation minimization
• Dual certificate
• Bessel-Vandermonde lift