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Vandermonde Matrices with Nodes in the Unit Disk and the Large Sieve
Information Theory · Submitted 10 Jan 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 Baz\'an'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 Baz\'an'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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.54 | 0.97 | 0.64 | asserted 1.00 |
| s1 | 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. | method 0.66 | 0.21 | 0.05 | asserted 1.00 |
| s2 | 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. | contribution 1.00 | 0.95 | 0.69 | asserted 1.00 |
| s3 | Compared to Baz\'an'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. | contribution 0.39 | 0.80 | 0.15 | asserted 0.98 |
| s4 | Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Baz\'an'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. | interpretation 0.40 | 0.79 | 0.05 | asserted 0.99 |
| s5 | As a byproduct, our result---when particularized to the case of nodes on the unit circle---slightly improves upon the Selberg-Moitra bound. | result 0.92 | 0.90 | 0.06 | asserted 0.99 |
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Original 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 Baz\'an'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 Baz\'an'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.
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Our arguments to prove Theorem [ref] are an adaptation of Selberg's argument for the large sieve (see [ref] ), along the lines of [ref] .
A planar large sieve and sparsity of time-frequency representations · Introduction · 0.91
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