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Real analysis: Integrable function

  1. May 10, 2009 #1
    1. The problem statement, all variables and given/known data
    HI all.

    In order to perform a Fourier transform on a function f(x), f(x) must be integrable, i.e.

    [tex]
    \int_{-\infty}^{\infty}|f(x)|dx < \infty.
    [/tex]

    Can you confirm that this also implies that f(x) -> 0 for x -> (+/-) infinity?
     
  2. jcsd
  3. May 10, 2009 #2

    Cyosis

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    Can you find a function that has a finite area beneath it while not being zero in infinity?
     
  4. May 10, 2009 #3
    No. But I am wondering if it is a sufficient conditions for the integral to be finite? E.g. f(x) = x-1?
     
  5. May 10, 2009 #4

    Cyosis

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    To avoid any confusion. Do you mean if f(x)->0 when x->+-infinity then the integral of f(x) over +- infinity converges? This is not true and you already gave a counter example. The function x^-1 goes to zero, but the integral of x^-1 from -infinity to infinity diverges.
     
  6. May 10, 2009 #5
    Great, thanks!
     
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