 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-69/3/2/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Combine the B-spline interpolation operator norm estimate from part 1 with the inverse operator norm computed in part 2(b):

∥ P_{x} ∥_{L^{\infty} \to L^{\infty}} \leq ∥ A_{x}^{- 1} ∥_{ℓ^{\infty} \to ℓ^{\infty}} < 2,

so in particular

∥ P_{x} ∥_{L^{\infty}} \leq 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.

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