The spectral properties of Vandermonde matrices with clustered nodes

We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of are “clustered” together – elements inside each cluster being separated by at most h?1N, clusters from other least ??1N. show that any pair column subspaces corresponding to two different nearly orthogonal: minimal principal angle between them is most?2?c1N??c2Nh, for constants c1,c2 depending only multiplicities clusters. As a result, spectral analysis VN significantly simplified reducing problem individually. Consequently we derive accurate estimates 1) all singular values V, 2) componentwise condition numbers linear squares problem. Importantly, these exponential local multiplicities, while changing linearly s.