Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3

The explicit divided difference formula makes $M_i(t)$ a finite linear combination of $(t_j-t{)}_{+}^{k-1}$. It is therefore polynomial of degree at most $k-1$ between knots and globally $C^{k-2}$. For $t<t_i$, the nodal data come from a polynomial of degree $k-1$, whose order-$k$ divided difference vanishes; for $t\ge t_{i+k}$ all truncated powers vanish. Thusand normalization does not change the degree, smoothness, knots, or support.

Apply the Leibniz rule for divided differences to $(x-t{)}_{+}^{k-1}=(x-t)(x-t{)}_{+}^{k-2}$. Only the zeroth and first divided differences of the linear factor survive. After applying the normalization, this gives the Cox-de Boor recursion formula

Induct on $k$, using the recursive divided-difference formula. After substituting the two induction hypotheses, useand the analogous identity for $g$; adjacent terms telescope. This proves