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Limit of integral lead to proof of convergence to dirac delta

  1. Feb 7, 2013 #1
    Hi,
    I try to prove, that function
    [itex]f_n = \frac{\sin{nx}}{\pi x}[/itex] converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if [itex] f_n [/itex]
    satisfy some conditions. So I need to show, that [itex]\lim_{n\rightarrow \infty}\int_{a}^{b}f_n \mathrm{d}x[/itex] is
    [itex]0[/itex] when [itex]0 [/itex] isn't in [itex] [a,b][/itex] and
    [itex]1[/itex] for [itex]0 [/itex] in [itex](a,b)[/itex] .

    I never met with problem before, the integral isn't "clasical" function and I don't have clue, how could I even start. I tried do some limit proceses, but it didn't show any concrete value - just estimation . . . (for other function which I found on wiki was possible count the integral and the limit is after easy . . .)

    Thaks for any help.
     
  2. jcsd
  3. Feb 7, 2013 #2

    mathman

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    Let y = nx and see what happens to the integral.
     
  4. Feb 8, 2013 #3
    Well, I get
    [itex] \lim_{n\rightarrow \infty} \int_{\frac{a}{n}}^{\frac{b}{n}} \frac{\sin{y}}{\pi y}dy [/itex]
    Right?
    Actually I tried this before, but what now? Thanks and sorry, if it is obvious, but I can't see it . .
    I tried do some limit processes, and if sign(a)=sign(b) and the function is there continues, that it
    is integral from "zero" to "zero" which could be zero . . . ? But what about other case, can't imagine the way I could get the result 1 . . .
     
    Last edited by a moderator: Feb 8, 2013
  5. Feb 8, 2013 #4

    mathman

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    You got the limits wrong. They should be axn and bxn, not a/n and b/n. The limit is just the infinite integral.
     
  6. Feb 8, 2013 #5
    You can't imagine, how did you help me! (sure, shame on me - i did just really stupid mistake) But integral from minus to plus infinity I can calculate, so problem solved!
    Thank you very much.
     
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