Finding polynomial optimal in terms of least squares approximation

Question

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that:

$$ (f,g) = \int\limits^1_0 f(x)g(x)dx. $$ Find $\|f-w\|$

Answer 1

The best approximation minimizing the norm is the projection $w$ onto the subspace spanned by the vectors $\{1,x,x^2\}$, which by the way, are not pairwise orthogonal in the given inner product.

We're looking for the $w$: 1) $w\in \operatorname{sp}(1,x,x^2)$, i.e., $w=w_1\cdot 1+w_2x+w_3x^2$, and 2) $(f-w)\perp \{1,x,x^2\}$.

Denoting $W = (w_1,w_2,w_3)^T$, $F=(<f,1>,<f,x>,<f,x^2>)^T$, and $G$ -- the Gramian matrix of the $\{1,x,x^2\}$, we should be able to solve for $W$ the resulting 3x3 matrix equation, $GW=F$.