Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 6 / a / Solution

Assume the spline knot sequence is ordered and its relevant support spans satisfy $t_{i+k}>t_i$. For distinct knots, the divided difference is a finite linear combination of evaluations, so it commutes with integration. With $u\in [a,b]$,Taking the order-$k$ divided difference in $u$ givesIndeed, the order-$k$ divided difference of a degree-$k$ polynomial is its leading coefficient, here $1/k$. This proves the unit-integral normalization of a B-spline. Admissible repeated knots follow by coalescing knots, with the usual one-sided endpoint conventions.