 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007/a/2/c/solution

Choose the nodes as the

n + 1

T_{n + 1}

. Its leading coefficient is

2^{n}

, so the monic nodal product is

ω = 2^{- n} T_{n + 1}

. Since

∣ T_{n + 1} ∣ \leq 1

[- 1, 1]

, part (b) bounds this particular interpolant by

∥ u - p_{n} ∥_{\infty} \leq \frac{∥ u ^{(n + 1)} ∥ _{\infty}}{2 ^{n} ( n + 1 )!} .

The best error cannot exceed the error of a particular member of

P_{n}

. Therefore

E_{n} \leq \frac{∥ u ^{(n + 1)} ∥ _{\infty}}{2 ^{n} ( n + 1 )!} .

The choice of Chebyshev roots controls the maximum of the nodal product, not the size of any individual interpolation coefficient.

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