The convolution of two function $f$ and $g$ is defined[1] as $$(f*g)(x) = \int f(y) g(x-y) dy.$$
Terry Tao explains very nicely on MathOverflow that the convolution with a bump function can be thought of as "blurring" the function. There is also a list of applications of convolutions on Wikipedia.
In these contexts, it appears the bump function used is usually symmetric about the $y$-axis, so it shouldn't matter whether we write $g(x-y)$ or $g(x+y)$. However, I have never seen the convolution defined with a plus sign: $$(f*g)(x) = \int f(y) g(x+y) dy.$$
If the point is to translate a bump function by $x$, then why is it standard to write $g(x-y)$ and not $g(x+y)$? Are there other uses of convolutions where it is important to write $g(x-y)$?
