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Real analysis: Integrable function

  • Thread starter Niles
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  • #1
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Homework Statement


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?
 

Answers and Replies

  • #2
Cyosis
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Can you find a function that has a finite area beneath it while not being zero in infinity?
 
  • #3
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Can you find a function that has a finite area beneath it while not being zero in infinity?
No. But I am wondering if it is a sufficient conditions for the integral to be finite? E.g. f(x) = x-1?
 
  • #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.
 
  • #5
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Great, thanks!
 

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