 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/11f/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Q_{n} \to f

L_{1}, \dots, L_{m}

x_{1}, \dots, x_{m}

, so

L_{j} (x_{i}) = δ_{ij}

P_{n} (x) = Q_{n} (x) + \sum_{j = 1}^{m} (f (x_{j}) - Q_{n} (x_{j})) L_{j} (x) .

Then

P_{n} (x_{j}) = f (x_{j})

for every

j

, while

∥ P_{n} - f ∥_{\infty} \leq ∥ Q_{n} - f ∥_{\infty} (1 + \sum_{j = 1}^{m} ∥ L_{j} ∥_{\infty}) ⟶ 0.

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