Hi,
If we define the convolution of two functions to be then this operation is supposed to commute, i.e. . When you change variables though, surely you get a minus sign???
Any help would be greatly appreciated.
Define . Then . But then the "limits of integration" are reversed in the 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 . Then its clear that if we consider the convolution and make the same change of variables, say , then we get . But then our limits of integration change positions: . So now, we flip the limits of integration to get
Thus, I believe a similar idea holds in the n dimensional case; so, in other words: .
I hope this helps!