We prove sharp lower bounds for the smallest singular value of a partial Fourier matrix with arbitrary "off the grid" nodes (equivalently, a rectangular Vandermonde matrix with the nodes on the unit circle), in the case when some of the nodes are separated by less than the inverse bandwidth. The bound is polynomial in the reciprocal of the so-called "super-resolution factor", while the exponent is controlled by the maximal number of nodes which are clustered together. As a corollary, we obtain sharp minimax bounds for the problem of sparse super-resolution on a grid under the partial clustering assumptions.

中文翻译：

---

具有集群节点的部分非均匀傅立叶矩阵的条件化

我们证明了具有任意“离网”节点（等效地，具有单位圆上的节点的矩形 Vandermonde 矩阵）的部分傅立叶矩阵的最小奇异值的尖锐下界，在某些节​​点被分隔的情况下小于反向带宽。该界限是所谓的“超分辨率因子”的倒数的多项式，而指数由聚集在一起的最大节点数控制。作为推论，我们在部分聚类假设下为网格上的稀疏超分辨率问题获得了尖锐的极小极大界限。