TITLE: What is convolution intuitively?
QUESTION [176 upvotes]: If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability distribution of $X+Y$.  This is the only intuition I have for what convolution means.
Are there any other intuitive models for the process of convolution?

REPLY [3 votes]: Although I like @TerryTao's answer, I would phrase a little differently. If you view $f$ as just a function and $g$ as a weight function with integral $1$ centered at $0$, then the convolution $f*g$ at $x$ is the weighted average of $f$ centered at $x$. This gives a "softened" approximation to $f$. If you rescale $g$ to maintain its integral at $1$ but concentrate it more and more at the origin, then it's clear the convolution will converge in some sene to $f$ itself. That's why it is often called an approximation of the identity.
If $f$ is a function of time, then the convolution of $f$ by $g$ can be viewed as a continuous analogue of a moving average of a sequence of measurements over time (except that the moving average usually has a time lag because it is using only values from the past). Both the moving average and convolution are used to soften the spikes that occur in the original sequence or function.