Vandermonde matrices with nodes in the unit disk and the large sieve

We derive bounds on the extremal singular values and the condition number of N×K, with N⩾K, Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link—first established by Selberg [1] and later extended by Moitra [2]—between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z∈C with |z|⩽1. Compared to Bazán's upper bound on the condition number [3], which, to the best of our knowledge, constitutes the only analytical result—available in the literature—on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Bazán's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result—when particularized to the case of nodes on the unit circle—slightly improves upon the Selberg–Moitra bound.