Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 70 / 5 / b

For $k=2$ on an equidistant spline knot sequence, $N_i$ is the triangular hat with value one at $t_{i+1}$ and support $[t_i,t_{i+2}]$. Its slopes on the two pieces are $1/h$ and $-1/h$, and $M_i=N_i/h$. Direct integration givesNonadjacent hats have disjoint interiors of their supports, so their inner product vanishes. Dividing by $h$ therefore givesThis linear-spline mixed Gram matrix is tridiagonal. Every diagonal entry is $2/3$, including the first and last: the distinct-knot hats have their full supports inside $[0,1]$. End rows simply lack one off-diagonal neighbor; there are no repeated endpoint spline knots here.