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We derive bounds on the extremal singular values and the condition number ofNxK, with N>=K, Vandermonde matrices with nodes in the unit disk. Themathematical techniques we develop to prove our main results are inspired by alink---first established by by Selberg [1] and later extended by Moitra[2]---between the extremal singular values of Vandermonde matrices with nodeson the unit circle and large sieve inequalities. Our main conceptualcontribution lies in establishing a connection between the extremal singularvalues of Vandermonde matrices with nodes in the unit disk and a novel largesieve inequality involving polynomials in z \in C with |z|<=1. Compared toBaz\'an's upper bound on the condition number [3], which, to the best of ourknowledge, constitutes the only analytical result---available in theliterature---on the condition number of Vandermonde matrices with nodes in theunit disk, our bound not only takes a much simpler form, but is also sharperfor certain node configurations. Moreover, the bound we obtain can be evaluatedconsistently in a numerically stable fashion, whereas the evaluation ofBaz\'an's bound requires the solution of a linear system of equations which hasthe same condition number as the Vandermonde matrix under consideration and cantherefore lead to numerical instability in practice. As a byproduct, ourresult---when particularized to the case of nodes on the unit circle---slightlyimproves upon the Selberg-Moitra bound.

Vandermonde Matrices with Nodes in the Unit Disk and the Large Sieve