Suppose that we have a function:
$\tilde U(\tau ,\omega ) = U(\tau ,\omega )f(\tau ,\omega ,Q)$
By evaluating this function, I would like to be able to find $Q$. Suppose that I don't know the explicit form of $U(\tau ,\omega )$ but I do know the explicit form of function $f(\tau ,\omega ,Q)$.
The $U(\tau ,\omega )$ and the $f(\tau ,\omega ,Q)$ can be complex-valued numbers. The $Q$ is a real integer.
Suppose that I evaluate the function a number of times (perhaps more times than shown here):
$\tilde U(\tau ,{\omega _0}) = U(\tau ,{\omega _0})f(\tau ,{\omega _0},Q)$
$\tilde U(\tau ,{\omega _1}) = U(\tau ,{\omega _1})f(\tau ,{\omega _1},Q)$
$\tilde U(\tau ,{\omega _2}) = U(\tau ,{\omega _2})f(\tau ,{\omega _2},Q)$
and so on, for as many function evaluations as required. When evaluating the function, I've fixed $\tau$ and $Q$ and I know $\omega_0$, $\omega_1$, $\omega_2$. The $Q$ remains constant for each function evaluation.
Is there a way (using numerical methods or non-linear curve-fitting) to find $Q$, only given $\tilde U(\tau ,\omega_0 )$, $\tilde U(\tau ,\omega_1 )$, $\tilde U(\tau ,\omega_2 )$ and $\omega_0$, $\omega_1$, $\omega_2$ and $\tau$ variables?
What might I be able to do in order to find Q? Might there be a way to change the problem so that I can find $Q$?