Thank you for pointing that out. It is assumed that $z$ is real. I edited the question.

In a research problem, I came across a class of kernels that are useful for a different reason. However, I'm trying to show that those same kernels produce solutions to this equation. Fedja: I think you pretty much answered my question. If I am understanding your answer, you are saying this problem definitely depends on the kernels themselves, given $f(x) = e^{-x}$. I am aware of techniques to answer my question for specific kernels. I just wanted to be sure there were no overarching theorems to cover my problem. Thank you!

Great point. Assume f is something like e^(-x).