Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 67 / 6 / b / Solution

The order-$k$ divided difference defining $M_i$ is a finite linear combination of values at its knots. Hence integration can be interchanged with it directly. For a knot value $u\in [a,b]$,It follows thatThe final equality holds because the order-$k$ divided difference of a degree-$k$ monic polynomial is its leading coefficient, one. ThusThis proves the unit-integral normalization of a B-spline. Its support is contained in $[t_i,t_{i+k}]$: below all its knots, the sampled truncated power agrees with a degree-$k-1$ polynomial, annihilated by the order-$k$ divided difference; above all its knots, every sample is zero. No mass is omitted by integration over $[a,b]$.

Since $N_i=(t_{i+k}-t_i)M_i/k$, the corresponding partition-normalized B-spline has integral $(t_{i+k}-t_i)/k$. Its normalization is different from that of $M_i$.