This question arises in the context of Cosmology, where the smoothing of a vector field $f(\vec{x})$ is defined through the following integration:
$$f_s(\vec{x})=\int d^3x'\ W(|\vec{x}-\vec{x}'|)f(\vec{x}')\tag{1}$$
with $W$ being a function called window function, usually a Gaussian. While reading research papers I noticed that the following notation is used:
$$\int d^3 x'\ W(|\vec{x}-\vec{x}'|)a(\vec{x}')b(\vec{x}')=a_sb_s\tag{2}$$
However, according to the definition, I believe that $a_sb_s$ should be:
$$a_sb_s=\bigg(\int d^3 x'\ W(|\vec{x}-\vec{x}'|)a(\vec{x}')\bigg)\bigg(\int d^3 x'\ W(|\vec{x}-\vec{x}'|)b(\vec{x}')\bigg)\tag{3}$$
How does this make sense? Is the integral in (2) equal, in general, to the product of integrals in (3)? Why?