 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-61/1/2/c/solution

For any degree-at-most-

n

t_{n}

f - v_{n, m} (f) = (f - t_{n}) - v_{n, m} (f - t_{n}) .

The operator norm bound in the preceding part yields

∥ f - v_{n, m} (f) ∥_{\infty} \leq (2 + \frac{2 n}{m}) ∥ f - t_{n} ∥_{\infty} \leq 2 (M + 1) ∥ f - t_{n} ∥_{\infty} .

∥ f - v_{n, m} (f) ∥_{\infty} \leq 2 (M + 1) E_{n} (f) .

No choice of a minimizer is needed for this argument. It is an instance of the polynomial reproduction error bound: a bounded linear reproducing operator has error at most

1 + ∥ P ∥

times the optimal error.

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