Consider the following convolution of a product of two functions $f(x)$ and $g(x)$:
$\int f(x')g(x')K_n(x-x') dx'$
where the kernel $K_n$ is a sequence of functions that approach a Dirac delta function as $n$ goes to $\infty$. In the limit as $n \rightarrow \infty$ we can write:
$\int f(x')g(x')K_{\infty}(x-x') dx' = \int f(x')K_{\infty}(x-x') dx' \int g(x')K_{\infty}(x-x') dx'$
that is this convolution of a product equals the product of two convolutions.
But for finite $n$ the convolution of a product is not in general equal to the product of convolutions. However for large $n$ we expect this to be approximately true. My question: How can one go about estimating the error in the following equation:
$\int f(x')g(x')K_n(x-x') dx' = \int f(x')K_n(x-x') dx' \int g(x')K_n(x-x') dx' + \epsilon_n(x)$.
I have faced this question in three different imaging scenarios this past year and I have found no treatment of the problem in the literature. It could be that I just don't know where or how to look.
Any help would be much appreciated,
DJS