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Dirac delta function as the limit of a seqquence

  1. Jan 19, 2010 #1
    Dirac delta function as the limit of a sequence

    Hi..

    If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x[tex]\neq[/tex]0, then can I call the limit as a dirac delta function?

    If not, what are the additional constraints to be satisfied?
     
    Last edited: Jan 19, 2010
  2. jcsd
  3. Jan 19, 2010 #2
    Let me refine that..

    If the integral of a member of the sequence over the whole real line is one, along with the conditions stated in previous post, can the limit of the sequence be identified with a dirac delta?

    Or are the conditions of the first post enough??
     
  4. Jan 19, 2010 #3

    Hurkyl

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    Let fn be any sequence of positive functions that converges (in the sense of distributions) to the dirac delta.

    Then 2fn does not converge to the dirac delta.

    Also, I believe [itex]\sqrt{f_n}[/itex] converges to 0.
     
  5. Jan 19, 2010 #4

    Hurkyl

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    Also,
    [tex]\lim_{n \rightarrow +\infty} g_n(x) = 0[/tex]​
    does not imply
    [tex]\lim_{n \rightarrow +\infty} g_n = 0[/tex]​
    (the second limit in the sense of distributions)

    In fact, I'm pretty sure you can find a sequence that satisfies the first equation whose limit is [itex]\delta(x-1)[/itex].
     
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