Use the orthogonality relation to find expressions for the coefficients $\displaystyle c_n$ in the eigenfunction expansion of a function f(x):

$\displaystyle f(x) \approx \sum_{n=1}^{\infty} c_n\phi_n$

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- May 4th 2010, 11:17 AMArythEigenfunction Expansion
Use the orthogonality relation to find expressions for the coefficients $\displaystyle c_n$ in the eigenfunction expansion of a function f(x):

$\displaystyle f(x) \approx \sum_{n=1}^{\infty} c_n\phi_n$ - May 4th 2010, 11:19 AMBruno J.
Hint : what is $\displaystyle <f, \phi_n>$?

- May 4th 2010, 11:40 AMAryth
Isn't it: $\displaystyle \int_a^b w(x)f(x)\phi_n(x)~dx$?

Sorry if I don't understand how it helps... I do know that $\displaystyle c_n$ is a quotient of inner products... I'm afraid I don't know how to show it. - May 4th 2010, 11:48 AMBruno J.
- May 4th 2010, 11:56 AMAryth
Ok... So... We get:

$\displaystyle <f,\phi_n(x)> = \sum_{n=1}^{\infty}c_n \int_a^b w(x)\phi_n(x)^2~dx = \sum_{n=1}^{\infty}c_n <\phi_n,\phi_n>$

Which means that:

$\displaystyle \frac{<f,\phi_n(x)>}{<\phi_n,\phi_n>} = \sum_{n=1}^{\infty} c_n$

Is that along the right lines? - May 4th 2010, 12:04 PMBruno J.
- May 4th 2010, 12:05 PMAryth
Ah, I see. I appreciate the help. Thanks a lot.

- May 4th 2010, 12:06 PMBruno J.