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I would like to apply Legendre polynomials to least square approximation. Therefore I would like the function:

$L_n (x)=\sum_{k=0}^n a_k P_k (x)$

to fit $f(x)$ defined over $[-1,1]$ in a least square sense.

We should minimize:

$I(a_0, ..., a_n)= \int_{-1}^1 [f(x) - L_n (x)]^2 \; dx\tag1$

and so we must set

$\frac{\partial I}{\partial a_r} = 0,\qquad r=0,1, \ldots,n\tag2$

Using equations $(1)$ and $(2)$

$\int_{-1}^1 P_r(x) \left[f(x) - \sum_{k=0}^n a_k P_k (x)\right]dx = 0,\qquad r=0,1, \ldots,n$

should be an equivalent term.

My question now is: why is that true?

I would be glad if someone could illustrate the last step with more details. Thanks, Rainier.

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    ok, equation (1) and (2) make perfect sense to me. The step which I don't understand is how to get to the last term.2012-09-15

3 Answers 3

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You have

$ \frac{\partial}{\partial a_r}\int_{-1}^1 [f(x) - L_n (x)]^2dx=0\\ \int_{-1}^1 \frac{\partial}{\partial a_r}[f(x) - L_n (x)]^2dx=0\\ \int_{-1}^1 2[f(x) - L_n (x)]\frac{\partial}{\partial a_r}[f(x) - L_n (x)]dx=0\\ \int_{-1}^1 2[f(x) - L_n (x)][0 - \frac{\partial}{\partial a_r}L_n (x)]dx=0\\ -\int_{-1}^1 2[f(x) - L_n (x)]\frac{\partial}{\partial a_r}\sum a_kP_k(x)dx=0\\ -\int_{-1}^1 2[f(x) - L_n (x)]P_r(x)dx=0 $

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It is even easier than that. Because the Legendre polynomials are orthogonal, you can get the coefficients just from $a_n=\frac {2n+1}2\int_{-1}^1f(x)P_n(x)dx$

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This is coming late but for the benefit of others, let me post my opinion:

$\frac{\partial\sum_{k=0}^n a_k P_k (x)}{\partial a_r} = \frac{\partial}{\partial a_r} [a_0 P_0 + a_1 P_1 + ... + a_r P_r + ... + a_k P_k] = P_r (x)$

This is because all other coefficients of $P_k (x) $ are treated as constants except for when $k=r$

The Orthogonality of the Legendre Polynomial was what was responsible for the final step

$\int_{-1}^1 P_r(x)P_k (x)dx = 0, \qquad r=0,1, \ldots,n$

The above equation is true as long as r is not equal to k