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. Such matrices arise in many fields of applied mathematics and engineering, e.g., in interpolation and approximation theory, sampling theory, compressed sensing, differential equations, control theory, and line spectral estimation. The mathematical techniques we develop to prove our main results are inspired by the 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. The Selberg-Moitra result employs Fourier-analytic techniques and the Poisson summation formula and therefore does not seem to be amenable to an extension to the case of nodes in the unit disk. 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| 6 1. This is accomplished by first recognizing that the Selberg–Moitra connection can alternatively be obtained based on the Montgomery–Vaughan proof technique for the large sieve, and then extending this alternative connection from the unit circle to the unit disk. 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 report can be consistently evaluated 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. This improved bound also applies to the square case, N = K, not covered by the Selberg–Moitra result.