Suppose that I have a function, $h$, which is the sum of several convolutions
$$ h = \sum_{i=1}^{n} f_{i}*g_{i} $$
where the asterisk represents convolution. By the convolution theorem we know that
$$ \mathcal{F}[f*g] = \mathcal{F}[f]\cdot\mathcal{F}[g] $$
where $\mathcal{F}$ is the Fourier transform and the dot represents elementwise multiplication. By the linearity of the Fourier transform, the first equation can be rewritten as
$$\mathcal{F}[h] = \sum_{i=1}^{n} \mathcal{F}[f_{i}]\cdot\mathcal{F}[g_{i}] $$
Now suppose that $h$ and $f_{1},f_{2},...,f_{n}$ are known and I want to infer $g_{1},g_{2},...,g_{n}$, or more realistically a least-squares approximation thereof.
If discretized, this problem bears some resemblance to a standard linear regression, but I haven't been able to come up with a form that lends itself to an analagous solution, and algebraic manipulation has only led me to dead ends—even after incorporating additional equations constraining the convolution relationships between each pair $f_{i}$ and $f_{j}$.
The major question that I haven't been able to answer is: does this problem have a solution? Or is it so under-constrained so as to be unsolvable?