Extra-large apertures, high carrier frequencies, and integrated sensing and communications (ISAC) are pushing array processing into the Fresnel region, where spherical wavefronts induce a range-dependent phase across the aperture. This curvature breaks the Fourier/Vandermonde structure behind classical subspace methods, and it is especially limiting with hybrid front-ends that provide only a small number of pilot measurements. Consequently, practical systems need continuous angle resolution and joint angle-range inference where many near-field approaches still rely on costly 2D gridding. We show that convexity can meet curvature via a lifted, gridless superresolution framework for near-field measurements. The key is a Bessel-Vandermonde factorization of the Fresnel-phase manifold that exposes a hidden Vandermonde structure in angle while isolating the range dependence into a compact coefficient map. Building on this, we introduce a lifting that maps each range bin and continuous angle to a structured rank-one atom, converting the nonlinear near-field model into a linear inverse problem over a row-sparse matrix. Recovery is posed as atomic-norm minimization and an explicit dual characterization via bounded trigonometric polynomials yields certificate-based localization that super-resolves off-grid angles and identifies active range bins. Simulations with strongly undersampled hybrid observations validate reliable joint angle-range recovery for next-generation wireless and ISAC systems.

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