Why two definitions of convolution are not equivalent?

Question

In half of the textbooks the convolution operation is defined as $$ (f * g)(x) = \int_{\mathbb{R}^d} f(x - y) g(y) dy $$ whereas in the other half, it is defined as $$ (f * g)(z) = \int_{\mathbb{R}^d} f(w) g(z - w) dw. $$ However it seems to me these two are not equivalent? Indeed in the first case, if I substitute $z = x - y$ then I get a minus in front since $dy = -dz$ $$ (f*g)(x) = - \int_{\mathbb{R}^d}f(z) g(x - z) dz $$

Answer 1

For an arbitary $d$ you make the substitution $w=x-y.$ The Jacobian matrix is equal $-I.$ The absolute value of its determinant is equal $1.$ Therefore you obtain $$(f*g)(x)=\int\limits_{\mathbb{R}^d} f(x-y)\,g(y)\,dy= \int\limits_{\mathbb{R}^d} f(w)\,g(x-w)\,dw$$

Answer 2

The two definitions are equivalent. Take $d = 1$ for the sake of simplicity. We have \begin{align} (f \star g)(x) &= \int_{-\infty}^\infty f(x - y) g(y)dy\\ &= -\int_{\infty}^{-\infty} f(z) g(x - z)dz\\ &= \int_{-\infty}^{\infty} f(z) g(x - z)dz \end{align}