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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 $\|f-w\|$

syntagma
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1 Answers1

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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, $GW=F$.

rych
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  • Hi Rych, i've been trying to understand best approximations under projections and this answer of yours is the only one I could find about. Please help me: when you say the best approximation, it's the best approximaiton to what? Also: If I project $x^3$ into the space spanned by $1,x,x^2$, then $f-w$ is the vector orthogonal to the subspace. I have to find the norm of this vector. Buw what is $w$? What it mean to be "optimal in terms linear squares approximation for a function $f(x) = x^3$"? – Guerlando OCs Apr 16 '19 at 21:00
  • Is $w$ the vector that minimizes $||f-w||$? – Guerlando OCs Apr 16 '19 at 21:12
  • Rych, please take a look at my question: https://math.stackexchange.com/questions/3183274/fitting-points-to-curve-gt-frac1001-alpha-e-beta-t-by-thinking-a I believe you know how to solve it, and it has a bounty – Guerlando OCs Apr 16 '19 at 21:14