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How would I construct a second degree Legendre Polynomial for $f(x)=cos(x)$ on the interval $[-1,1]$? I am not understanding the explanation in the book. I just want to know how to start.


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What do you mean by "a Legendre Polynomial for $f(x)$"? Do you mean a partial sum of the Fourier-Legendre series? Exactly your example is done at – Robert Israel Nov 5 '12 at 19:40
Sorry, I meant a second degree Legendre least squares approximation to $f(x)=cos(x)$. Would I it be the same thing? – Alti Nov 5 '12 at 19:43
Yes, the partial sums of the Fourier-Legendre series are the least squares approximations by linear combinations of Legendre polynomials. – Robert Israel Nov 5 '12 at 21:44
Please change the title, since that is not what you look for. If you wish you might even add the answer yourself. – AD. Aug 19 '13 at 16:31

A second order polynomial can be uniquely defined with three distinct points (in general, an order $n$ polynomial requires $n+1$ points).

What does $\cos x$ look like on $[-1,1]$? Well, $\cos$ is symmetric about the $y$-axis. So let's use $x = -1, 0, 1$ as our interpolating points. Then, we wish to find a polynomial such that $P(-1) = \cos (-1)$, $P(0) = \cos 0$, $P(1) = \cos 1$.

Then, it depends on how you're defining the Legendre polynomials. There is a common definition for the Legendre family of orthogonal polynomials, but I'm not entirely sure this is what you are looking for.

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