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Math Help - Eigenfunction Expansion

  1. #1
    Super Member Aryth's Avatar
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    Eigenfunction Expansion

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

    f(x) \approx \sum_{n=1}^{\infty} c_n\phi_n
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    MHF Contributor Bruno J.'s Avatar
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    Hint : what is <f, \phi_n>?
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    Super Member Aryth's Avatar
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    Isn't it: \int_a^b w(x)f(x)\phi_n(x)~dx?

    Sorry if I don't understand how it helps... I do know that c_n is a quotient of inner products... I'm afraid I don't know how to show it.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Aryth View Post
    Isn't it: \int_a^b w(x)f(x)\phi_n(x)~dx?

    Sorry if I don't understand how it helps... I do know that c_n is a quotient of inner products... I'm afraid I don't know how to show it.
    Well that's the left side of the equation, indeed. But what if you replace f by its expansion in the expression <f, \phi_n>, and then use the linearity of the inner product?
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    Super Member Aryth's Avatar
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    Ok... So... We get:

    <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:

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

    Is that along the right lines?
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    MHF Contributor Bruno J.'s Avatar
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    Be careful! You're using n both as a fixed integer and as a summation index! Here's what your corrected post looks like :

    Quote Originally Posted by Aryth View Post
    Ok... So... We get:

    <f,\phi_n(x)> = \sum_{j=1}^{\infty}c_j <\phi_j,\phi_n> = c_n <\phi_n, \phi_n>
    because all the terms cancel (by orthogonality), except <\phi_n, \phi_n>.
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  7. #7
    Super Member Aryth's Avatar
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    Ah, I see. I appreciate the help. Thanks a lot.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Aryth View Post
    Ah, I see. I appreciate the help. Thanks a lot.
    Most welcome! Good luck.
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