Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$,
$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$
Is it possible to solve for the integral kernel, $K(\mathbf{w,\mathbf{v}})$, if $f(\mathbf{w})$ and $g(\mathbf{v})$ are known scalar functions and we require
$\int K(\mathbf{w,\mathbf{v}}) \, d\mathbf{v} = 1 \, ?$
Follow-up note: these are definite integrals, $\int \rightarrow \int_{a1}^{b1} \int_{a2}^{b2} \int_{a3}^{b3}$
Thank you for any insight.