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.

翻译：超大孔径、高载波频率以及集成感知与通信（ISAC）技术正将阵列处理推向菲涅尔区域，其中球面波前会在孔径上引发随距离变化的相位。这种曲率破坏了经典子空间方法所依赖的傅里叶/范德蒙结构，在仅能提供少量导频测量的混合前端系统中尤为受限。因此，实际系统需要连续角度分辨率及联合角度-距离推断，而现有许多近场方法仍依赖计算成本高昂的二维网格化处理。本文证明，通过一种基于提升的无网格超分辨率框架，凸性可与曲率特性相结合以处理近场测量。其核心在于对菲涅尔相位流形进行贝塞尔-范德蒙分解，从而揭示角度维度中隐藏的范德蒙结构，同时将距离依赖性隔离至紧凑的系数映射中。基于此，我们引入一种提升映射，将每个距离单元与连续角度对应至一个结构化秩一原子，从而将非线性近场模型转化为对行稀疏矩阵的线性逆问题求解。恢复问题被构建为原子范数最小化，并通过有界三角多项式给出显式对偶表征，进而实现基于可验证定位的超分辨率离网角度估计与活跃距离单元识别。在强欠采样的混合观测场景下的仿真实验验证了该方法在下一代无线通信及ISAC系统中实现可靠联合角度-距离恢复的有效性。