Given the equation:

\begin{gather*} f(t) = \int_{\Omega }k(x)g(t, x)\,dx \\ \\ 0 \leq t \leq ∞ \\ a \leq x \leq b & a,b \in \mathbb{R} \end{gather*}

If $k(x)$ is an unknown function but $f(t)$ and $g(t,x)$ are known - Can one bring $k(x)$ out of the integral to potentially solve for it?

Thank you!

  • 1
    $\begingroup$ What are the domains/spaces for each of the functions? $\endgroup$ – Shine On You Crazy Diamond Oct 26 '15 at 2:07

There are cases, e.g. for a point charge in electrodynamics, where one can find an operator $L$ for which $g$ is a Green's function, then $$ L f(t) = \int\limits_\Omega k(x) \, L \, g(x,t) \, dx = \int\limits_\Omega k(x) \, \delta(x-t) \, dx = k(t) $$

  • $\begingroup$ That is interesting because the initial condition at $ g(0,x)= \delta(x-x0) $ $\endgroup$ – Edv Beq Oct 26 '15 at 10:49

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