MATHEMATICS OF COMPUTATION Volume 66, Number 220, October 1997, Pages 1555–1560 S 0025-5718(97)00897-1 B-SPLINES AND OPTIMAL STABILITY J.M. PE ˜ NA Abstract. It is proved that, among all nonnegative bases of its space, the B-spline basis is optimally stable for evaluating spline functions. 1. Introduction In some recent papers [3]–[5], several optimal properties of the B-spline basis have been studied. Different viewpoints have been considered; for instance, the shape preserving properties in Computer Aided Geometric Design (see [4]) or the supports of the basis functions (see [3]). The B-spline basis b =(b 0 ,...,b n ) is a normalized nonnegative basis, that is b i ≥ 0 ∀ i =0,...,n, and ∑ n i=0 b i = 1. The interest of normalized nonnegative bases of a space comes from their convex hull property. In Computer Aided Geometric Design, this property implies that, for any control polygon, the corresponding curve always lies in the convex hull of the control polygon. In this paper, we shall prove a property of the optimal stability of the B-spline basis among all nonnegative bases of its space. Given a basis u =(u 0 ,... ,u n ) of a real vector space U of functions defined on Ω and a function f ∈ U , there exists a unique sequence of real coefficients (c 0 ,...,c n ) such that f (t)= n  i=0 c i u i (t) for all t ∈ Ω. One practical aspect to consider in the evaluation of the function f is the stability with respect to perturbations of the coefficients, which depends on the chosen basis of the space. We want to know how sensitive a value f (t) is to random perturbations of a given maximum relative magnitude ε in the coefficients c 0 ,...,c n corresponding to the basis. Following [7] and [6], we can bound the corresponding perturbation δf (t) of the change of f (t) by means of a condition number C u (f (t)) := n  i=0 |c i u i (t)|, for the evaluation of f (t) in the basis u: |δf (t)|≤ C u (f (t))ε. Received by the editor May 10, 1995 and, in revised form, July 29, 1996. 1991 Mathematics Subject Classification. Primary 65D07, 41A15. Key words and phrases. B-splines; optimal stability, condition number, nonnegative matrices, partial ordering. This work was partially supported by the Spanish Research Grant DGICYT PB93-0310 and by the EU project CHRX-CT94-0522. c 1997 American Mathematical Society 1555 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use