Spectral Conditioning of Weighted Multivariate Vandermonde Operators via Radial Beurling–Selberg Approximation

Joseph Gabet, Maxime Ferreira Da Costa, and Kiryung Lee

Abstract

We establish explicit eigenvalue bounds for Gram matrices of weighted multivariate Vandermonde operators with separated nodes on a torus. The squared magnitudes of the weights are samples of a nonnegative radial spectral energy density whose profile has bounded variation, finite energy, and finite weighted variation cost; compact support is not required. The estimate holds in every dimension and depends on the periodic Euclidean separation and the weighted variation integral. For compactly supported profiles, our analysis combines a signed layer decomposition with the ball approximants of Holt and Vaaler; the noncompactly supported case follows by truncation. Combining the resulting radial envelopes with a bandlimited sampling identity based on Poisson summation yields the spectral bounds by eliminating off-diagonal interactions and nonzero diagonal aliases. The result covers nonmonotone profiles and discontinuous radial low-pass filters, and it is complemented by a global elementary upper bound on the Holt-Vaaler cost that exhibits two radial regimes.

Cite

@misc{gabet:hal-05862059,
  author     = {Gabet, Joseph and Ferreira Da Costa, Maxime and Lee, Kiryung},
  title      = {Spectral Conditioning of Weighted Multivariate {Vandermonde} Operators via Radial {Beurling--Selberg} Approximation},
  year       = {2026},
  eprint     = {hal-05862059},
  eprinttype = {hal}
}