Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 69 / 3 / 2 / c / Solution

Combine the B-spline interpolation operator norm estimate from part 1 with the inverse operator norm computed in part 2(b):so in particular $\Vert P_x{\Vert }_{L^{\infty }}\le 2$, uniformly in $n$. The subpartition of unity for B-splines is important at the outer boundary: the finite basis may sum to less than one there, but that only strengthens the estimate.