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Math Help - how to find the inner product formulla

  1. #1
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    how to find the inner product formulla

    once i solve that one is the derivative of the other

    but here its much harder to guess the formulla
    http://i47.tinypic.com/ixt74i.jpg



    what is the general method?
    Last edited by transgalactic; November 21st 2009 at 03:08 AM.
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  2. #2
    Senior Member Shanks's Avatar
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    Fourier series

    It is suffice to prove the following statement:
    \lim_{n\to \infty} \int_{0}^{\pi} f(x)\sin{nx}=0 \text { and }\lim_{n\to \infty} \int_{0}^{\pi} f(x)\cos{nx}=0
    These conclusion follows immediately from the Fourier series in  L^{2}\text{ space}.
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  3. #3
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    i want to find the formula for the inner product which defines such norm.

    what it has to do with proving that its foorier coefficients
    ?
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  4. #4
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    The Trigonometric function system is a complete orthonormal base for the Hilbert Space L^{2} .
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  5. #5
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    i am looking for answer like
    <f,g>=\int f(x)g(x) + etc..

    an actual formula

    i cant translate your words into such formula
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  6. #6
    Senior Member Shanks's Avatar
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    To prove the first statement : Extend the domain of f to [-\pi ,\pi] such that f is odd function.
    To prove the second statement : Extend the domain of f to [-\pi ,\pi] such that f is even function.
    The inner product is defined as:
    < f , g >=\int_{-\pi}^{\pi} f *g dx
    in both case.
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  7. #7
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    what prove?

    i dont ask to prove anything

    i ask how to find the formula which defines this norm
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  8. #8
    Senior Member Shanks's Avatar
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    norm ? sorry , there may be some mistakes , I didn't see any norm in your question and picture.
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  9. #9
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    ohh sorry its the wrong foto
    i will change it in a moment
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  10. #10
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    http://i47.tinypic.com/ixt74i.jpg

    a little change

    "which defines the minimal"
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  11. #11
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    i need to find alpha beta and gama
    so this expression will be minimal

    i know how to solve such stuff
    usually
    i have a vector and a subspace to make a projection of the vector

    so i make an orthogonal basis and then i make a projection of that vector
    into my space

    and then the difference between that vector and the original vector is the minimal

    but in order to do all that i need the
    inner product formula which defines this norm.

    usually i figured out the formula by guessing

    but here i cant guess

    so i am asking if there is a general method
    ?
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  12. #12
    Senior Member Shanks's Avatar
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    Product Hilbert space

    Since \mathbb{C} and L^{2} are both complex Hilbert Space,
    Then \mathbb{C}\times L^{2} is also Hilbert Space under the inner product defined by:
    < (r,f(x)) , (s,g(x)) >= r*\overline{s}+\int_{0}^{1}f(x)\overline{g(x)}dx
    The norm is induced from the inner product.
    The minimal problem is equivalent to find the distance between point (1,3x^2) and The closed subspace spanned by (1,1) and (1,2x) and (1,0).
    Indeed, Since \alpha is a free variable, you can eliminate the |1-(\alpha+\beta+\gamma)|^{2} part, it doesn't affect the final minimal value (but it has something to do with minimal point).
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  13. #13
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    how??

    what is (r,f(x)) ?

    what is (s,g(x))?

    what is s? what is r?
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  14. #14
    Senior Member Shanks's Avatar
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    Here \mathbb{C}\times L^2 is the cartesian product of \mathbb{C} (the complex number field) and L^2, and r , s are any complex numers, f(x) and g(x) are any elements of L^2.
    Do you know what is Cartesian Product?
    Last edited by Shanks; November 21st 2009 at 05:26 AM. Reason: Latex error
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  15. #15
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    its the sum of the multiplication on coordinates with the same index.

    how to get the formula from your definition
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