 Solution

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.