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Proving a given sequence is a delta sequence ~

  1. Oct 4, 2006 #1
    Hi! I'm in a mathematical ecology class and we're working with delta sequences.

    I'm trying to show that

    delta_n(x) = n if |x| <= 1/2n
    = 0 if |x| > 1/2n

    is a delta sequence.

    ----Definition of a delta sequence---------------------------------------
    Suppose delta_n is a sequence of functions with the property that

    lim (int delta_n*f(x) dx, -infinity, infinity) = f(0)
    n->inf

    for all smooth, absolutely integrable functions f(x). Then delta_n is a delta sequence.
    -----------------------------------------------------------------------

    I thought I could start it by inserting the function into the definition, breaking up the resulting integral, and taking some limits (after some integration by parts, possibly) -- but that's gotten me nowhere. Perhaps I'm missing something there?

    A "hint" that comes with the problem is:
    Apply the Mean Value Theorem to a function of the form
    F(x) = int( f(t) dt, a, x).

    I'm not even sure how this hint applies.

    I've been racking my brain a couple days and would totally appreciate some guidance! :)

    Respectfully,

    Michael
     
  2. jcsd
  3. Oct 4, 2006 #2

    Galileo

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    Since f is a nice function, let F be an antiderivative of f. Then you can just evaluate the integral in terms of F (and n). Now you need to evaluate the limit and end up with f(0) somehow.
     
    Last edited: Oct 4, 2006
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