Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 2 / a

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- Solution a

Solution

The Chebyshev alternation theorem says that

p^{*} \in P_{n}

is the unique best uniform approximation to

f \in C [- 1, 1]

if and only if there are at least

n + 2

ordered points

- 1 \leq t_{1} < t_{2} < \dots < t_{n + 2} \leq 1

ε \in {- 1, 1}

such that

f (t_{i}) - p^{*} (t_{i}) = ε (- 1)^{i} ∥ f - p^{*} ∥_{\infty}, i = 1, \dots, n + 2 .

Thus the extremal error has common magnitude and alternating sign at

n + 2

points.

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