Assume $K \in \mathbb{R} $ and $f \in L ^1 (\mathbb{R})$. Suppose the fourier transform of a function $g$ and its inverse exist. Find an integral form of the solution of $$f(t) = g(t) + K[g(t-1) + g(t+1)]$$ and give a condition on $K$ which makes the solution valid.
From being asked to find a condition on the constant $K$ to see where the solution is valid, this type of question reminds me of how we solve differential equations: anytime we solve a differential equation, we should also be concerned about the domain of validity. However, I do not see how differential equations could be incorporated between the equation given and the Fourier transform. Does anyone have an idea of how to solve this?
Taking the Fourier Transform, $$\hat{f}(k) = \hat{g} (k) + K[ e^{-2\pi i k} \hat{g} (k) + e^{2\pi i k} \hat{g} (k) ],$$ but I do not see how this leads anywhere.