 Solution

Apply the general B-spline interpolation operator norm estimate in part (I) to the inverse norm from (b):and hence in particular $\Vert P_{\mathbf{x}}{\Vert }_{L^{\infty }}\le 3$, independently of $n$. The strict finite-dimensional estimate uses the actual finite collocation matrix; it does not require periodic closure, natural boundary conditions or endpoint knot repetitions.

There is also a direct shorter verification of the uniform bound. For an index $i$ maximizing $|z_i|$, the triangle inequality giveswith an even larger margin when a neighbor is missing. This is the inverse infinity-norm bound from diagonal dominance, and again gives $\Vert A_n^{-1}{\Vert }_{{\ell }^{\infty }}\le 3$.