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 can, in the worst case, grow slightly stronger than exponentially with the spatial dimension. Both results improve over all previously known results of such type and cannot be improved considerably which is shown by considering specific node sets.

我们证明了最小奇异值的显式下界和复单位圆上具有分散节点的矩形多元 Vandermonde 矩阵的条件数的上界。类似于香农-奈奎斯特准则，假设节点被一个常数除以使用的多项式次数分隔。如果这个常数随空间维度线性增长，则条件数是一致有界的。如果它仅随空间维度呈对数增长，则在最坏的情况下，条件数的增长可能比随空间维度呈指数增长略强。这两个结果都比以前已知的所有此类结果都改进了，并且不能通过考虑特定节点集来显着改进。