Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 61 / 3

Write the strictly increasing local spline knots as $u_j=t_{i+j}$, $0\le j\le k$. The explicit formula for a divided difference givesEach summand is a truncated power function of $t$: it is a polynomial on either side of its knot $u_j$, and for $k\ge 2$ is globally $C^{k-2}$. Hence $M_i$ is a piecewise polynomial function of degree at most $k-1$, with these spline knots and at least this global smoothness.

For $t\ge u_k$, all summands vanish. For $t<u_0$, all knot values agree with those of the ordinary polynomial $(s-t{)}^{k-1}$ in the divided-difference variable $s$. Its order-$k$ divided difference is zero, because its degree is less than $k$. Thus $M_i$ also vanishes to the left of $u_0$.

The closed support is exactly the indicated interval, rather than merely contained in it. For $u_0<t<u_1$, subtracting the omitted $j=0$ term from the zero divided difference of $(s-t{)}^{k-1}$ givesFor $u_{k-1}<t<u_k$, only the last truncated-power term survives, givingThere are therefore nonzero values arbitrarily close to either endpoint, provingAt each simple knot, the $(k-1)$st derivative has a nonzero jump from exactly one truncated-power summand, so this is also the exact global smoothness. For order $k=1$, the function is instead the normalized interval indicator; there is no assertion of classical continuity, and the notation $C^{k-2}$ is only the customary formal spline smoothness notation. This distinction is part of simple-knot B-spline regularity.

We derive the Cox-de Boor recurrence from the Leibniz rule for divided differences. Fix $t$, let $u_j=t_{i+j}$, and setAway from the spline knots, $h(s)=(s-t{)}_{+}^{k-1}$. IntroduceThe defining divided-difference recurrence says $(u_k-u_0)D=R-L$. Since $s-t=(s-u_0)+(u_0-t)$ and the first factor is linear, the Leibniz rule for divided differences givesMultiplication by $u_k-u_0$ consequently yieldsReplace $L$ and $R$ by the corresponding lower-order B-splines divided by their support widths. This provesThe denominators are positive for the distinct spline knots of this question. The recursion starts with $N_{i,1}$, the interval indicator on $[t_i,t_{i+1})$. For $k\ge 2$, values at the spline knots follow by the continuous extension of the left side; the usual half-open convention for the order-one factors gives the same result. This proves the recurrence rather than assuming it as a definition.