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So I have to find the following $\min_{a,b,c\in\mathbb{R}}\int_{-1}^{1} |x^3-a-bx-cx^2|^2dx$

I have a hint at a solution which says to consider $X=\{\mbox{polynomials of degree} \leq 2\}$.

So then we have $\min_{a,b,c\in\mathbb{R}}\int_{-1}^{1} |x^3-a-bx-cx^2|^2dx=\inf_{g\in X} ||x^3-g||$ for some $g\in X$

Where the norm $||.||$ is defined using the inner product $=\int_{-1}^{1} f(x)\bar{g(x)}dx$

So then I think I'm supposed to use an orthogonal projection somehow I think (and maybe find some orthonormal basis?) but I'm a bit lost as to how to do any of this.

Thanks for any help.

 For completeness I am putting my solution (from the answers below). 

The problem reduces to finding a basis for $X$ and then orthonarmalizing it. We then use the fact that the orthohgonal projection onto the space $X$ will give the minimal distance to it so we need to calculate: $||x^3-P(x^3)||$ where $P(x^3)=\sum_{i=1}^{3} e_i$ for an orthonormal basis $\{e_i\}$.

Noting that the set $\{1,x,x^2\}$ spans the space $X$ we then apply the gram-schmidt process to this set to give a set of orthogonal vectors: $\{1,x,(x^2-\frac{1}{3})\}$.

This basis now needs to be normalized but if we notice that in :

$\sum_{i=1}^{3} e_i$

The first and third terms will cancel as the integrand will be odd, so we only need to normailze the middle vector which gives $\{\sqrt{\frac{3}{2}\}}$.

And so $P(x^3)=\sqrt{\frac{3}{2}}x=\frac{3}{5}x$

So we now have $||x^3-\frac{3x}{5}||=\sqrt{\int_{-1}^{1} (x^3-\frac{3x}{5})^2dx}=\sqrt{\frac{8}{175}}$

Which is the desired distance

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    @Alex yeah thanks sorry2012-10-23

2 Answers 2

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In fact you need to find square of the distance from the vector $x^3$ to the subspace spanned by $1,x,x^2$ in innner product space of polynomials on $[-1,1]$.

Here is a big picture approach. Let $H$ be a inner product space, $X$ its finite dimensional subspace with orthogonal basis $\{e_1,\ldots,e_n\}$. Let $x\in H$, then the orthogonal projection of $x$ onto $X$ is given by $ \mathrm{pr}_X(x)=\sum\limits_{k=1}^n \langle x,e_k\rangle e_k\tag{1} $ The dsired distance is $ \mathrm{dist}(x,E)=\Vert x-\mathrm{pr}_E(x)\Vert\tag{2} $ In your case take $H$ to be the space of all polynomials on $[-1,1]$, and $X$ to be the space of all polynomials of degree less than $2$. In order to apply formula $(2)$ you need orthogoanl basis of $X$. To get it apply Hilbert-Schmidt orthogonalization process to vectors $1,x,x^2$.

So it is remains to perform all calculations.

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    Such a nice answer! +12018-12-02
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$ \begin{eqnarray} \min f\left(a,b,c\right) &=& \min\int_{-1}^1 dx \left(x^3 - a - bx - cx^2\right)^2 \\ &=& 2 \min \left[\left(a^2 + \frac{2}{3}ac + \frac{1}{5}c^2\right) + \left(\frac{1}{3}b^2 - \frac{2}{5}b\right) + \frac{1}{7}\right] \\ &=& 2 \min \left[g\left(a,c\right) + h\left(b\right) + \frac{1}{7}\right] \\ &=& 2 \left[g\left(0,0\right) + h\left(- \frac{3}{25}\right) + \frac{1}{7}\right] \\ &=& \frac{8}{175}. \end{eqnarray} $

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    You added the square root at the end in your solution. The square root does not exist in the original integral.2012-10-24