How to sum a function using its DFT?

Question

I have a function $f(x_n)$ defined over $n$ points, which is initially unknown, but its DFT is known as $F(k_n)$. However, I need to compute $\sum_n f(x_n)$ from the DFT. How to accomplish this without the inverse transform, just using the Fourier modes?

Note: The $n$ point grid is known a priori and is uniformly sampled over $[0, L]$.

Answer 1

$\textbf{Hint}$: The DFT is another summation involving $f$

$$F(k_n) = \sum_m f(x_m)\exp(j k_n x_m)$$

Can you think of a way to make the exponential vanish?