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How to correctly convolve two delta functions?

  1. Jun 17, 2015 #1
    How do I correctly compute the convolution of two delta functions? For example, if I want to compute ##\delta(\omega)\otimes\delta(\omega)##, I should integrate $$\int_{-\infty}^\infty \delta(\omega-\Omega)\delta(\Omega) d\Omega$$
    This integrand "fires" at two places: ##\Omega = 0## and ##\Omega = \omega##, and evaluates to ##\delta(\omega)## in either case. That leads me to say that ##\delta(\omega)\otimes\delta(\omega) = 2\delta(\omega)##, which somehow feels wrong... I want it to be just ##\delta(\omega)##.

    Anyone know the correct treatment here?
     
  2. jcsd
  3. Jun 17, 2015 #2

    RUber

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    In your example, ##\int_{-\infty}^\infty \delta(\omega-\Omega) \delta(\Omega) \, d\Omega## is only non-zero if both ## \omega - \Omega = 0 ## and ## \omega = 0 ##.
    So for this, you end up with ##\delta(\omega)##.
     
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