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Dirac delta approximation - need an outline of a simple and routine proof

  1. Jul 22, 2008 #1
    Hi, I need your help with a very standard proof, I'll be happy if you give me some detailed outline - the neccessary steps I must follow with some extra clues so that I'm not lost the moment I start - and I'll hopefully finish it myself. I am disappointed that I can't proof this all by myself, but I really need to move on with my work so any help here will be appreciated.

    Let [tex]\theta[/tex] be a function that decays faster than any polynomial, is smooth or whatever else is required. Let

    [tex]\int_{-\infty}^{\infty} \theta = K[/tex]

    I want to show, that for all reasonable [tex]f[/tex] (continuous, smooth, bounded, ... again - whatever is required) the following identity holds

    [tex]\lim_{a \rightarrow 0}\int_{-\infty}^{+\infty} f(x) \frac{1}{a}\theta\left(\frac{x}{a}\right) dx = K f(0) [/tex]

    In other words - the limit

    [tex]\lim_{a \rightarrow 0} \frac{1}{a}\theta\left(\frac{x}{a}\right)[/tex]

    is a multiple of Dirac delta, in the sense of distributions. The proof needs not to be 100% precise, even 80% precise, I need it for some little physics paper I write and physicists are seldom 100% precise (in math, anyway), but I need something a little more rigorous than "Let's assume [tex]\theta[/tex] is rectengular..." for which the proof is way too easy.

    I appreciate any help, thanks in advance. This is no hw, if it's any important.
  2. jcsd
  3. Jul 22, 2008 #2


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    What lines of attack have you tried on this? It's hard to give helpful hints without knowing where you're at on the problem!

    Anyways, two high-level observations that are almost surely useful to the proof (or at least to the devising of a line of attack) are

    [tex]\int_{-\infty}^{+\infty} \theta \approx \int_{-H}^{+H} \theta[/tex]


    [tex]f(\epsilon) \approx f(0),[/itex]

    capturing the notions that [itex]\theta[/itex] is rapidly decaying and that f is continuous at 0, respectively.

    (H is a 'large' positive number, [itex]\epsilon[/itex] is a 'small' positive number)

    (I use the integral expressing that the tails are irrelevant, rather than something more direct like [itex]\theta(\pm H) \approx 0[/itex], because the integral actually appears in the data you've given)
  4. Jul 22, 2008 #3
    Are you sure you don't want [itex]\frac{1}{a}\theta (a x)[/itex]?
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