论文标题
汇合Vandermonde矩阵的广义尖峰和光谱的超分辨率
Super-resolution of generalized spikes and spectra of confluent Vandermonde matrices
论文作者
论文摘要
我们研究了嘈杂的傅立叶测量值在一维圆圈上狄拉克分布及其衍生物的线性组合的超分辨率的问题。在有关该主题的最新著作之后，我们考虑了“部分聚类”的几何设置，当时某些狄拉克可以在瑞利限制以下的地方分开。在这个假设下，我们证明了相应的矩形汇合vandermonde矩阵的最小奇异值的尖锐渐近界限，并在单位圆上带有节点。结果，在属于固定网格的节点的附加假设下，我们得出了上述超分辨率问题的下部和上部最大最大误差边界的匹配。
We study the problem of super-resolution of a linear combination of Dirac distributions and their derivatives on a one-dimensional circle from noisy Fourier measurements. Following numerous recent works on the subject, we consider the geometric setting of "partial clustering", when some Diracs can be separated much below the Rayleigh limit. Under this assumption, we prove sharp asymptotic bounds for the smallest singular value of a corresponding rectangular confluent Vandermonde matrix with nodes on the unit circle. As a consequence, we derive matching lower and upper min-max error bounds for the above super-resolution problem, under the additional assumption of nodes belonging to a fixed grid.
