I have a function f(x) defined below as,
$f(x) = \sum_{n=0}^N c_n(g_n(x + n) + g_n(x - n))$
I have no knowledge of what $c_n$ is, but I do have access to f(x) numerically. Is there a way to apply an operator that would cancel all terms except for one associated with a particular n? My goal is to extract $g_n(x)$. A few notes, $g_n(0) = 1$, $g_n(x) = 0$ as $|x|$ becomes large.
I have tried a handful of things such as moving to the fourier domain where terms such as $g_n(x+n) + g_n(x-n)$ are equivalent to the fourier representation $g_n(k)\cos(nx)$. But I am struggling to develop a filter, numerical or analytical, that is able to attenuate or filter out $g_m(x)$ where $n\neq m$.
If we let $\bar{\mathbf{g}}(x) = \mathbf{g}(-x)$, then I think what you have written is equal to
$$ \mathbf{f} = \mathbf{c} \ast \mathbf{g} + \mathbf{c} \ast \bar{\mathbf{g}} = \mathbf{c} \ast ( \mathbf{g} + \bar{\mathbf{g}}). $$
Witht knowing neither $\mathbf{c}$ nor $\mathbf{g}$, this is a blind deconvolution problem, which may or may not be solvable depending on the characteristics of both $\mathbf{c}$ and $\mathbf{g}$.