# Can I solve for a unique integral kernel?

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.

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Isn't this related to the identification of time varying channels (regularize.wordpress.com/tag/time-varying-channels)? –  Dirk Sep 18 '12 at 21:09
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. –  user31373 Sep 18 '12 at 23:17