Title: Vandermonde Matrices with Nodes in the Unit Disk and the Large Sieve,
Abstract: We derive bounds on the extremal singular values and the condition number of
NxK, 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 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 \in 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.