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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.

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    @ Thomas- my above question is the real core of the post. Can we do a small modification on that $e^{ikx}$ function to take advantage of fourier modes. Tell me what you think in the answers section please. I have been trying my best on this.2011-12-30

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