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Math Help - Orthogonal to orthonormal

  1. #1
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    Orthogonal to orthonormal

    I hava a orthogonal basis {1, t, 3t^2-1 , 5t^3-3t}

    and this normalized is

    {1,t, 1/2(3t^2-1), 1/2(5t^3-3t)}

    I dont understand how they have done this. Any help would be much appreciated. Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by adam_leeds View Post
    I hava a orthogonal basis {1, t, 3t^2-1 , 5t^3-3t}

    and this normalized is

    {1,t, 1/2(3t^2-1), 1/2(5t^3-3t)}

    I dont understand how they have done this. Any help would be much appreciated. Thanks.
    You should tell us what interval (a,b) these are supposed to be orthonormal over.

    The usual norm for real functions over such an interval is:

    \|f\|_{_2}=\left[ <br />
\int_a^b f(x)^2\;dx<br />
\right]^{1/2}

    If you have not made any typos I could work out what interval is implied by the given normalisations, but then you could just tell us.


    CB
    Last edited by CaptainBlack; February 21st 2010 at 05:23 AM.
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  3. #3
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    In any case, once you have an orthogonal set of vectors, you can "normalize" them by dividing each vector by its length.

    Apparently, your interval, [a, b], is such that "1" and "t" already have length 1 while " 3t^2- 1" and " 5t^3- 3t" both have length 1/2.
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  4. #4
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    interval 1 and -1
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  5. #5
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    Quote Originally Posted by adam_leeds View Post
    interval 1 and -1
    Then we already have a problem since:

    \left[\int_{-1}^1 1^2\;dt\right]^{1/2}=\sqrt{2}

    so normalising f(t)=1 gives \widehat{f}(t)=\frac{f(t)}{\|f\|_2}=1/\|1\|_2=1/\sqrt{2}

    CB
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  6. #6
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    Quote Originally Posted by adam_leeds View Post
    I hava a orthogonal basis {1, t, 3t^2-1 , 5t^3-3t}

    and this normalized is

    {1,t, 1/2(3t^2-1), 1/2(5t^3-3t)}

    I dont understand how they have done this. Any help would be much appreciated. Thanks.
    In the case of these Legendre polynomials, the purpose of that "normalisation" is not to make them orthonormal, but to ensure that they have the value 1 when t = 1. See that Wikipedia link for further information.
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