Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 7 / b / Solution

For distinct ordered knots, put $h_i=t_{i+1}-t_i>0$. The linear B-spline $N_i$ is the triangular hat supported on $[t_i,t_{i+2}]$, equal to $(t-t_i)/h_i$ on its rising segment and $(t_{i+2}-t)/h_{i+1}$ on its falling segment. Its integral-normalized partner is $M_i=2N_i/(h_i+h_{i+1})$. ThusFor example, the left overlap is ${\int }_0^{h_i}(u/h_i)(1-u/h_i)du=h_i/6$. Nonadjacent hats have disjoint interiors of their supports, so their inner product is zero. The linear-spline mixed Gram matrix therefore has rowRetain only column indices in $1,\dots ,n$: the first or last row has no neighbor outside this range. Admissible repeated endpoint knots are obtained by a limit with zero spacings, provided $h_i+h_{i+1}>0$; a degenerate hat with zero support width is not a basis element.