 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-71/4/2/c/solution

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

The general estimate from part 1 and the exact inverse norm from part 2b imply

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

and in particular $∥ P_{x} ∥_{L^{\infty}} \leq 2$ , uniformly in the number of knots. This bound concerns the support-centred sites

x_{i} = i + 3/2

; shifting the sampling sites changes the B-spline collocation matrix and can destroy uniform stability.

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