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Thread: Riemann integrable

  1. #1
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    Riemann integrable

    Good morning to you all,

    any genius in the house ? does anybody know how to solve this ?? please help. thank you

    Ps: f_k is f sub k
    f^ is f hat
    f^(n) = nth Fourier coeffcient of f

    suppose that {f_k } is a sequence where k goes from 0 to infinity, is a sequence of Riemann integrable functions on the interval [0; 1] such that

    INTEGRAL FROM 0 TO 1 OF |f_k (x) - f(x)| dx -> 0 as k -> infinity

    Show that f^_k(n) -> f^(n) uniformly in n as k -> infinity
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  2. #2
    Super Member girdav's Avatar
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    Quote Originally Posted by PeaceSoul View Post
    suppose that $\displaystyle \left\{f_k\right\}_{k\geq 0}$ is a sequence of Riemann integrable functions on the interval [0; 1] such that

    $\displaystyle \lim_{k\to+\infty}\int_0^1|f_k (x) - f(x)| dx = 0$

    Show that $\displaystyle \lim_{k\to+\infty}\hat{f_k}(n) =\hat{f}(n)$ uniformly in $\displaystyle n$.
    What is the definition of $\displaystyle \hat f(n)$?
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  3. #3
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    ^
    f(n) which is the nth fourier coefficient of f

    L is the length of the interval so L= b-a

    interval is [a,b]

    ^
    f (n) = (1/L ). integral from a to b of { f(x).e^ ( ( -2(pi)inx )/L )}dx

    here is a better representation

    http://s1138.photobucket.com/albums/...rrent=fhat.jpg
    thank you for ur reply
    Last edited by PeaceSoul; May 8th 2011 at 07:32 AM.
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  4. #4
    Super Member girdav's Avatar
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    We only have information about the behavior on $\displaystyle (0,1)$ whereas $\displaystyle \widehat f(n)$ needs to know the integral on $\displaystyle (a,b)$.
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  5. #5
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    kindly go to this page to see a better represenation of f hat.. note that the integral is from a to be

    http://s1138.photobucket.com/albums/...rrent=fhat.jpg
    Last edited by PeaceSoul; May 8th 2011 at 07:31 AM.
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