Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 4

Put $m=k-1$, so the Marsden identity isDifferentiating $m-j$ times with respect to $x$ givesThe exact Taylor formula for a polynomial isSubstitution of the differentiated Marsden identities yieldswhere

Differentiating this expression for ${\lambda }_i$ produces two sums whose adjacent terms cancel. The uncancelled endpoints contain ${\psi }_i^{(k)}$ and $p^{(k)}$, both zero because the two functions have degree at most $k-1$. Thusso the Marsden dual functional ${\lambda }_i(p)$ is independent of the auxiliary point $x$.

Fix ${\xi }_i\in [t_i,t_{i+k}]$ and choose a knot interval $I=(t_{\ell },t_{\ell +1})$ adjacent to it on which $N_i$ is active. Exactly $k$ B-splines are nonzero on $I$, and their polynomial restrictions form a basis of ${\mathcal{P}}_{k-1}$. Applying the expansion from part (a) to the polynomial piece $N_j{|}_I$ givesUniqueness of coordinates in this local basis forceswhen $N_j$ is active. If $N_j$ vanishes on $I$, all of its local polynomial derivatives vanish and the same equality holds with value zero. At a knot, use either adjacent polynomial piece; the $x$-independence proved in part (a) gives the same coefficient. Henceso the ${\lambda }_i$ form the dual basis to the B-spline basis.