 Solution

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

T_{6}

T_{6} (x) = 32 x^{6} - 48 x^{4} + 18 x^{2} - 1.

Subtract its monic version from

x^{6}

p (x) = x^{6} - \frac{1}{32} T_{6} (x) = \frac{3}{2} x^{4} - \frac{9}{16} x^{2} + \frac{1}{32} .

This has degree four and hence belongs to the usual space

P_{5}

. The error is

T_{6} /32

, with norm

1/32

and alternating extrema at the seven points

cos (jπ /6)

j = 0, \dots, 6

. Part (a), followed by uniqueness, proves

∥ x^{6} - p ∥_{\infty} = \frac{1}{32}

and that this is the unique best approximation of degree at most five.

If “degree five” is taken literally to exclude degree four, no best approximation exists in that nonclosed class:

p + ϵ x^{5}

approaches the error

1/32

ϵ \to 0

P_{5}

excludes attainment by any degree-five polynomial. This is why the standard at-most-degree convention matters here.

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