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Thread: Operation of convoulution is commutative

  1. #1
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    Operation of convoulution is commutative

    Hi,

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

    Any help would be greatly appreciated.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by markwolfson16900 View Post
    Hi,

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

    Any help would be greatly appreciated.
    Define $\displaystyle t=x-y$. Then $\displaystyle \,dy=-\,dt$. But then the "limits of integration" are reversed in the $\displaystyle \displaystyle\int_{\mathbb{R}^n}$ 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 $\displaystyle \mathbb{R}$. Then its clear that if we consider the convolution $\displaystyle f*g(t) = \displaystyle\int_0^t f(t-\tau)g(\tau)\,d\tau$ and make the same change of variables, say $\displaystyle s=t-\tau$, then we get $\displaystyle \,d\tau=-\,ds$. But then our limits of integration change positions: $\displaystyle \displaystyle\int_{t}^{0}f(s)g(t-s)(-\,ds)$. So now, we flip the limits of integration to get $\displaystyle \displaystyle\int_0^t f(s)g(t-s)\,ds=g*f(t)$

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

    I hope this helps!
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