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.
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!