A Discrete Abel-Hilbert Principle for Conditioning of Weighted and First-Order Confluent Vandermonde Operators

Joseph Gabet and Maxime Ferreira Da Costa

Abstract

We establish explicit eigenvalue bounds for the Gram matrices associated with two families of structured Vandermonde-type operators involving discrete weights and separated nodes on the torus. The first family covers weighted Vandermonde operators, for which we obtain bounds in terms of the node separation and the discrete variation of the weights. The second family concerns first-order confluent extensions under a symmetry assumption on the weights. In this confluent setting, we prove a strongly decoupled bound: the final estimate depends only on the diagonal contributions of the two blocks and involves no cross terms. Our analysis is entirely discrete. It relies on a summation-by-parts argument of Abel type, combined with a Hilbert-type inequality for the associated sampled kernel matrices.

Cite

@misc{gabet:hal-05667282,
  author     = {Gabet, Joseph and Ferreira Da Costa, Maxime},
  title      = {A Discrete {Abel-Hilbert} Principle for Conditioning of Weighted and First-Order Confluent {Vandermonde} Operators},
  year       = {2026},
  eprint     = {hal-05667282},
  eprinttype = {hal}
}