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 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.

Bespoke extensional elasticity through helical lattice systems

“…As our final result for this section, we present, for later use, a bound on the singular values of the square Vandermonde matrix V J ( ψ 1 , ψ 2 , …, ψ J+1 ) defined in (5.6 b ). This immediately follows, for the case ψ 1 , ψ 2 , …, ψ J+1 ∈(0, 1], from the result in [29, section 3.2].…”

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

“…Vandermonde matrices with complex nodes appear in polynomial interpolation problems and many other fields of mathematics (see, e.g., the introduction of [2] and its references). In this paper, we are interested in rectangular Vandermonde matrices with nodes on the complex unit circle and with a large polynomial degree.…”

“…The condition number of those matrices has recently become important in the context of stability analysis of super-resolution algorithms like Prony's method [6,15], the matrix pencil method [12,18], the ESPRIT algorithm [20,21], and the MUSIC algorithm [17,22]. If the nodes of such a Vandermonde matrix are all well-separated, with minimal separation distance greater than the inverse bandwidth, bounds on the condition number are established for example in [2,5,14,18].…”

Conditioning of Partial Nonuniform Fourier Matrices with Clustered Nodes

“…, s. Denote ∆ N := min i =j |ξ i,N − ξ j,N |. Several more or less equivalent bounds on σ min (V N ) are available in the "well-separated" case N ∆ N > const, using various results from analysis and number theory such as Ingham and Hilbert inequalities, large sieve inequalities and Selberg's majorants [23,30,34,3,31,32,19,9].…”

“…Vandermonde matrices and their spectral properties are of considerable interest in several fields, such as polynomial interpolation, approximation theory, numerical analysis, applied harmonic analysis, line spectrum estimation, exponential data fitting and others (e.g. [3,5,9,36,37,39] and references therein). Motivated by questions related to the so-called problem of super-resolution (more on this in Subsection 3.2 below), in this paper we study the conditioning of rectangular Vandermonde matrices V with irregularly spaced nodes on the unit circle, where the number of nodes s is considered to be relatively small and fixed, while the polynomial degree N ≥ s can be large.…”