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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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    Sorry, I wrote it wrong... the basis is different from the other one :)2012-09-01

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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.