---
abstract: |
    The famous de Boor conjecture states that the condition of the polynomial B-spline collocation matrix at the knot averages is bounded independently of the knot sequence, i.e., depends only on the spline degree.

    For highly nonuniform knot meshes, like geometric meshes, the conjecture is known to be false. As an effort towards finding an answer for uniform meshes, we investigate the spectral condition number of cardinal B-spline collocation matrices. Numerical testing strongly suggests that the conjecture is true for cardinal B-splines.
author:
- 'Vedran Novaković [^1], Sanja Singer [^2] and Saša Singer [^3]'
title: 'Estimates for the Spectral Condition Number of Cardinal B-Spline Collocation Matrices (Long version)[^4]'
---

[**Keywords**]{}: [cardinal splines, collocation matrices, condition, Töplitz matrices, circulants]{}

[**AMS subject classifications**]{}: [65D07, 65D05, 65F35, 15A12]{}

Introduction
============

We consider the classical Lagrange function interpolation problem in the following “discrete” setting. Let $\tau_1, \ldots, \tau_\nu$ be a given set of mutually distinct interpolation nodes, and let $f_1, \ldots, f_\nu$ be a given set of “basis” functions.

For any given function $g$ we seek a linear combination of the basis functions that interpolates $g$ at all interpolation nodes, $$\sum_{j = 1}^{\nu} y_j f_j(\tau_i) = g(\tau_i), \quad i = 1, \ldots, 