On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Understand

We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle.

• The nodes are "off the grid", pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance.
• Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes.
• For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.