Results 1 to 2 of 2

Math Help - Operation of convoulution is commutative

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    8

    Operation of convoulution is commutative

    Hi,

    If we define the convolution of two functions f,g:\mathbb{R}^n \longrightarrow \mathbb{R} to be f* g(x) = \int_{\mathbb{R}^n} f(x-y)g(y)dy then this operation is supposed to commute, i.e. f* g = g* f. When you change variables though, surely you get a minus sign???

    Any help would be greatly appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by markwolfson16900 View Post
    Hi,

    If we define the convolution of two functions f,g:\mathbb{R}^n \longrightarrow \mathbb{R} to be f* g(x) = \int_{\mathbb{R}^n} f(x-y)g(y)dy then this operation is supposed to commute, i.e. f* g = g* f. When you change variables though, surely you get a minus sign???

    Any help would be greatly appreciated.
    Define t=x-y. Then \,dy=-\,dt. But then the "limits of integration" are reversed in the \displaystyle\int_{\mathbb{R}^n} part. So to get them in the right order, we "flip" the limits and then make that result negative.

    Like for instance, consider the case we're integrating over \mathbb{R}. Then its clear that if we consider the convolution f*g(t) = \displaystyle\int_0^t f(t-\tau)g(\tau)\,d\tau and make the same change of variables, say s=t-\tau, then we get \,d\tau=-\,ds. But then our limits of integration change positions: \displaystyle\int_{t}^{0}f(s)g(t-s)(-\,ds). So now, we flip the limits of integration to get \displaystyle\int_0^t f(s)g(t-s)\,ds=g*f(t)

    Thus, I believe a similar idea holds in the n dimensional case; so, in other words: \displaystyle\int_{\mathbb{R}^n}f(x-y)g(y)\,dy \xrightarrow{t=x-y}{} -\int_{\mathbb{R}^n}f(t)g(x-t)(-\,dt)=\int_{\mathbb{R}^n}f(t)g(x-t)\,dt=g*f.

    I hope this helps!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. row operation
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 8th 2011, 02:19 PM
  2. commutative operation
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: July 31st 2010, 04:22 AM
  3. Operation @
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: April 2nd 2010, 12:10 AM
  4. Operation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 24th 2010, 08:20 PM
  5. Commutative Operation 71.10
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 28th 2009, 11:06 AM

Search Tags


/mathhelpforum @mathhelpforum