I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int e^{ikx}h(x)dx\right)\left(\int_{-\infty}^{\infty}e^{ikx}\,\bar{f(k')}dk'-\int_{\mathbb{R}}e^{-ikx}h(x)dx\right).$$
I want to choose the appropriate $f$ in terms of $h$ to make this above expression go to zero. I want to use Fourier series. Also, a little clarification on notation: in the integral, $k$ is fixed for now, since we are summing over all the $k$, as seen on the far left. I am trying to approach this by bring $e^{\pm ikx}$ out of the respective integrals where we are integrating with respect to the variable $k'$ and then proceeding. But that leads to some pathologies when trying to make this term go to zero.