Another Hilbert inequality and critically separated interpolation nodes

Abstract: Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.


Introduction and known results
For m, n ∈ N, z j ∈ C, j = 1, . . . , m, we define the Vandermonde matrix where q := min j̸ =ℓ min r∈Z |t j − t ℓ + r|. An approach via extremal functions [5] or via a generalized Hilbert inequality [1] then yields Obviously, this bound detoriates for nq → 1 although full row rank m is still guaranteed via a Vandermonde determinant argument since m ≤ 1/q ≤ n. Moreover, for nq < 1 and n large enough, there can be placed m > n equispaced nodes on the unit cirlce and thus the matrix A cannot have rank m.

Specific setup and main result
In the following we discuss one particular critically separated case nq = 1 and show that unlike for the Fourier matrix, the condition number grows with n. Let n ∈ N, q = 1 2n+1 , m = 2n, and nodes be given by x j = jq, j = 1, . . . , n, j + 1 2 q, j = n + 1, . . . , 2n, see Figure 1 for an illustration. Now, we define the Vandermonde matrix A n := e 2πikxj j=1,...,2n;k=−n,...,n ∈ C 2n×2n+1 and the symmetric Toeplitz matrix The following two lemmata discuss the close relationship of these matrices and approximate eigenvectors of the matrix T n . Lemma 2.1 With the above notation, the extremal singular values of A n and the maximal eigenvalue of T n are related via and since the A n has full rank 2n, the matrix K n is symmetric positive definite. A theorem of Jordan-Wielandt [4, Thm. 7.3.3] yields the result.
P r o o f. Please note that the non-numbered constant C may differ at each appearance. Only considering the first half of the vector and estimating sin x ≤ x, we directly compute Subsequently, we bound w ⊤ (I n − T n )w from above, this involves several preparatory steps: The definition of the vector v is motivated by the continuous analogue where the first equality uses t = tan x and s = tan y and the second equality follows from t = 1 2 z + z −1 and Cauchy's integral formula. For fixed y ∈ (0, π 2 ), the integrand S y : (0, π 2 ) → R, is convex and with h = π/(2n + 1) and j = 1, . . . , n, we obtain where the last inequality follows by leaving out the minimal summand and moving the remaining ones inwards. Together with and an analogous estimate for the integral on the interval [π/2 − h/2, π/2], this yields . Together with sin x ≥ 2x/π, L := diag(ℓ), and using the symmetry of the vectors w = Lv and v, we get Secondly, there exists κ ∈ (0, 1) such that To see this, we start by noticing the symmetry of ℓ and v as well as cos x ≥ cos(π/2 − x), 0 ≤ x ≤ π/4, such that .
for i, j < n/2. We get for j ≤ ⌈n/2⌉ the bound Moreover, the subsequent summands are all negative and can be estimated by where the first inequality either adds positive or neglects negative terms, respectively. Combining both estimates leads to n j which is non-positive for log(4j/n) ≤ −1 and thus shows the inequality (1). Together with sin x ≥ 2x/π and again the symmetry of w and v we get www.gamm-proceedings.com Theorem 2.3 There exist a constant C 0 > 0 such that for sufficiently large n ∈ N and with the above notation, we have and in particular cond A n ≥ C 0 log n.
P r o o f. The upper bound follows from Lemma 2.1 and symmetry of T n by 0 < σ 2 min A n = 1 − λ max T n = 1 − ∥T n ∥. The lower bound follows from Lemma 2.2 by Finally, we would like to mention the following similar result for the Hilbert matrix. We have where the upper bound might be seen as a variant of the famous Hilbert inequality and the lower bound can be found in [3]. While we have the very same rate for the second term, our constant certainly is an artefact of our proof technique. Unfortunately, we were not able to relate T n and H n directly. The numerical test in Figure 2 suggests a "true" constant C = 4/π 2 in Theorem 2.3. An with respect to n (blue stars), numerical value of 1 − w ⊤ w/w ⊤ T w for the test vector w (red stars), and 1 + 4 π 2 log(n) (yellow line). Note that the norm of the matrix Tn can be computed explicitly for moderate n and via Matlab's routine svds, where we used a fast matrix vector multiplication via fast Fourier transforms, for large n.