Spline functions of negative degree (Q1590726)

Given two sets of knots, \(\{v_j\}\) and \(\{u_j\}\), such that \(v_1\leq v_2\leq\dots\leq v_n\leq\dots u_1\leq u_2\dots\), the B-splines of negative degree \(-n\) are defined by \[ N_{k,-n}(x)=\begin{cases} \frac{x-v_{k+n}}{u_k-v_{k+n}} N_{k-1,-n-1}(x)+\frac{u_{k+1}-x}{u_{k+1}-v_{k+1+n}}, & k\geq 0\\ 0, & k<0.\end{cases} \] The purpose of this paper is to obtain new representations for B-splines of degree \(-n\). In addition, the structure of the space \(B^{(n)}_m\) spanned by \(\{N_{0,-n},\dots,N_{m,-n}\}\), is investigated. The following result is proved: Theorem. The space \(B^{(1)}_m\) is an \((m+1)\)-dimensional Haar subspace of \(C(-\infty,u_1)\). For the set of knots \(K=\{t_1,\dots,t_N\}\), a definition of the class of spline functions of negative degree is presented. This class, which is denoted by \(S^{(n)}(K)\), is formed by some type of piecewise rational functions. Some results and remarks on the class \(S^{(n)}(K)\) are given.