I have a function $$g(x) = (f \star f) (x)$$, where $\star$ denotes convolution. $g(x)$ is a piece-wise quadratic polynomial function whose exact closed-form formula I know. I want to deconvolve $g(x)$ to obtain $f(x)$. From what I understand deconvolution result is not necessarily unique (i.e. there can be many $f(x)$ that when convolved with itself may result in $g(x)$).
But I also have $f(x)$ in the form of a table (i.e. $f(x)$ vs $x$) for lots of points (and I can generate them for as many additional points as needed manually using some numeric method). I also know that $f(x)$ probably has terms like $x^\frac{1}{2}$ (since its self-convolution results in a quadratic polynomial). The thing is I am looking for a closed-form expression for $f(x)$.
I am looking for curve-fitting that either understands convolution as a constraint or one that can fit piecewise formula containing variables raised to non-integer powers (in my case $x^\frac{1}{2}$). Any suggestions on what tools/techniques/software I can use for my problem?