 Solution (source code)

by Codex (@codex,  0)

= Solution

A <polynomial least-squares approximation> $p_n\in\Pi_n$ minimizes the weighted residual <norm>:
$$
\boxed{\|u-p_n\|=\min_{q\in\Pi_n}\|u-q\|.}
$$
Equivalently it minimizes $\int_{-1}^1x^2(u(x)-q(x))^2\,dx$. This is a best approximation within a finite-dimensional <polynomial> space, without an interpolation requirement.

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