B-splines and optimal stability

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) greater than or equal to O For All i = 0,..., n, and Sigma(i=0)(n) 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 Omega and a function f epsilon U, there exists a unique sequence of real coefficients (c(0),..., c(n)) such that f(t) = Sigma(i=0)(n)c(i)u(i)(t) for all t epsilon Omega . 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 epsilon in the coefficients c(0),..., c(n) corresponding to the basis. Following [7] and [6], we can bound the corresponding perturbation Sf(t) of the change of f(t) by means of a condition number C-u(f(t)):= Sigma(i=0)(n)\c(i)u(i)(t)\, for the evaluation of f(t) in the basis u: \delta f(t)\ less than or equal to C-u(f(t))epsilon.