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I'm currently trying to solve an equation of the form $$f(x) = \sum_m\,a_m\,\varphi_m(x)$$ and it's required me to project this equation on a different set of functions $$\{\phi_m(x)\}$$ that is orthonormal on the interval (a,b).

How do I execute such kind of projection?

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The question is not clear. Your $f$ is already a linear combination of $\{\varphi_m\}$. Can you make your question more clear? –  Siminore Sep 1 '12 at 9:58
    
Sorry, I wrote it wrong... the basis is different from the other one :) –  Juan Sebastian Totero Sep 1 '12 at 10:42
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up vote 1 down vote accepted

I think the coefficients of $f$ on the new basis should be given by

\begin{align} a'_n=\int_a^b f(x)\phi_n(x)w(x)dx &=\int_a^b \sum_m a_m\varphi_m(x)\phi_n(x)w(x)dx\\ &=\sum_m a_m\int_a^b \varphi_m(x)\phi_n(x)w(x)dx\\ \end{align}

supposing the integral-sum exchange is possible and where $w(x)$ is the weight function of the given scalar product.

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