TITLE: In what sense the convolution formula is described as a weighted average?
QUESTION [5 upvotes]: In this Wikipedia article about convolution, it is said that: 
$$
(f*g)(t):=\int_{-\infty}^{+\infty} f(\tau)g(t-\tau)d\tau
$$
can be described as a weighted average of the function $f(\tau)$ at the moment $t$. I don't understand what the "weighted average" means here. I've seen this again and again whenever convolution is mentioned. According to another article about weighted functions, the "average" should be:
$$
\frac{\int_{-\infty}^{+\infty} f(\tau)g(t-\tau)d\tau}{\int_{-\infty}^{+\infty} g(t-\tau)d\tau}.
$$
Could someone clarify it?

REPLY [3 votes]: Some of the most important convolutions (with heat kernel or Poisson kernel) involve nonnegative functions $g$ with $\int g=1$, in which case weighted average is an accurate description. In general, weighted sum would be better. Then again, it's still a weighted average (if $g\ge 0$), up to a constant factor. In a context where a constant factor is not important    (e.g., one is interested in asymptotic behavior, or in the smoothness of $f*g$)  it is not surprising that the "weighted average" / "weighted sum" distinction is blurred.