3
$\begingroup$

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.

  • 1
    Not enough information. Knowing one input vector and one output vector, you cannot tell what linear transformation is at work. Consider the discrete version of this problem, where $g$, $f$, $K$ are piecewise constant and the integrals become sums - you will see that the problem is underdetermined.2012-09-18

0 Answers 0