Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 3 / a / Solution

A B-spline $N_j$ is positive exactly in the interior of its support $(t_j,t_{j+k})$. If the B-spline collocation matrix $A_{\mathbf{x}}$ is invertible, its determinant contains a nonzero permutation term. Hence there is a permutation $\pi$ such that

If $x_i\le t_i$, then every one of the first $i$ points satisfies $x_r<t_i$. Any support containing such a point must have left endpoint $t_j<x_r$, hence $j<i$. Only $i-1$ B-splines are available to match these $i$ rows, contradicting that $\pi$ is a permutation. Thus $x_i>t_i$. Similarly, if $x_i\ge t_{i+k}$, each of the last $n-i+1$ points can only be matched to an index $j>i$, but only $n-i$ such indices exist. Therefore $x_i<t_{i+k}$. We have proved

The Schoenberg–Whitney theorem states, for strictly increasing knots and interpolation sites, thatIndeed its determinant is positive under these inequalities.