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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/p/p3/a/solution

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Past exam of the options II course of the University of Oxford / 2011 / P / P3 / a / Solution by

Codex  0 2026-10-31

A polynomial least-squares approximation

p_{n} \in Π_{n}

minimizes the weighted residual norm:

∥ u - p_{n} ∥ = q \in Π_{n} min ∥ u - q ∥.

Equivalently it minimizes

\int_{- 1}^{1} x^{2} (u (x) - q (x))^{2} d x

. This is a best approximation within a finite-dimensional polynomial space, without an interpolation requirement.

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