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Math Help - Fourier Series/Dirichlet Conditions

  1. #1
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    Fourier Series/Dirichlet Conditions

    Dirichlet conditions



    -------------------------------------

    1. Ok, after plotting/sketching the curves, I think f(x) doesnt violate any dirichlet conditions, and g(x) does. But im not sure which condition g(x) violates?

    2. How would i represent f(x) a fourier series? integrate 1/x?
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  2. #2
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    Quote Originally Posted by gomes View Post
    Dirichlet conditions



    -------------------------------------

    1. Ok, after plotting/sketching the curves, I think f(x) doesnt violate any dirichlet conditions, and g(x) does. But im not sure which condition g(x) violates?

    2. How would i represent f(x) a fourier series? integrate 1/x?
    You need the periodic extensions of $$ f and $$g , and you need the conditions for a fundamental period of 1 not 2\pi.

    Given that, here you should be looking at the absolute integrability condition.

    CB
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  3. #3
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    thanks, lets say the conditions instead of 2pi or pi etc..are replaced by 1?
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  4. #4
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    Quote Originally Posted by gomes View Post
    thanks, lets say the conditions instead of 2pi or pi etc..are replaced by 1?
    Which of:

    \displaystyle \int_0^1 |f(x)|\; dx <\infty

    and

    \displaystyle \int_0^1 |g(x)|\; dx <\infty

    are true?

    CB
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  5. #5
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    Thanks! the integral of g(x) is true, but the integral of f(x) is false.

    So that means g(x) can be represented as a fourier series and f(x) cant.

    Question: So how do i show with calculations that g(x) can be represented as fourier series and f(x) cant? do i just use the formulae below (and use the limits 0 and 1, instead of pi)?

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  6. #6
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    Quote Originally Posted by gomes View Post
    Thanks! the integral of g(x) is true, but the integral of f(x) is false.

    So that means g(x) can be represented as a fourier series and f(x) cant.

    Question: So how do i show with calculations that g(x) can be represented as fourier series and f(x) cant? do i just use the formulae below (and use the limits 0 and 1, instead of pi)?

    When you adjust these to have the correct fundamental yes, but note (in this case it is obvious as both $$ f and $$ g are positive functions, but would be true in general if the absolute integrability condition fails) $$ a_0 only exists for $$ g(x)

    CB
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