在单位圆上具有分离结点的多元范德蒙矩阵是稳定的
Multivariate Vandermonde matrices with separated nodes on the unit circle are stable
摘要：
我们证明了明确的最小奇异值的下界和上界的条件数的矩形，多元Vandermonde矩阵与分散的节点上的复杂单位圆。类似于Shannon-Nyquist准则，假设节点由常数除以所使用的多项式次数分开。如果这个常数随空间维数线性增长，则条件数是一致有界的。如果它仅随空间维度呈指数增长，则条件数随空间维度的增长略强于呈指数增长。这两个结果是准最优的，并改善了所有以前已知的结果，这种类型。
英文摘要：
We prove explicit lower bounds for the smallest singular value and upper bounds for the condition number of rectangular, multivariate Vandermonde matrices with scattered nodes on the complex unit circle. Analogously to the Shannon-Nyquist criterion, the nodes are assumed to be separated by a constant divided by the used polynomial degree. If this constant grows linearly with the spatial dimension, the condition number is uniformly bounded. If it grows only logarithmically with the spatial dimension, the condition number grows slightly stronger than exponentially with the spatial dimension. Both results are quasi optimal and improve over all previously known results of such type.
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