Another Hilbert inequality and critically separated interpolation nodes

Abstract Abstract: Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.