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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-69/6/a/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

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]

,

\int_{a}^{b} (u - t)_{+}^{k - 1} d t = \int_{a}^{u} (u - t)^{k - 1} d t = \frac{( u - a ) ^{k}}{k} .

Taking the order-

k

divided difference in

u

gives

\int_{a}^{b} M_{i} (t) d t = k [t_{i}, \dots, t_{i + k}] \frac{( u - a ) ^{k}}{k} = 1 .

Indeed, 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.

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