Why discrete fourier transform consider only complex coefficients with non-negative indices

Question

This is surely a silly question.

$f(x)$ for $x \in [0, 2\pi]$

$f(x)=\sum_{k=-\infty}^{\infty} c_k e^{i kx}$

However, when we consider the discrete fourier transform, we seem to estimate $f(x)$ with $\sum_{k=0}^{N} c_k e^{i kx}$, instead of $\sum_{k=-N}^{N} c_k e^{i kx}$

I think the reason should related to $[0, 2\pi]$, since its on one side, but I couldn't figure it out.

Please help.

Answer 1

$$\exp[2\pi i kx/N]=\exp[2\pi i (N-(N-k))x/N]=\exp[-2\pi i(N-k)x/N] , \ x\in Integers$$

So there is a redundancy between $k$ and $-(N-k)$, which effectively just doubles $\exp[2\pi i kx/N]$. This only works if $x$ is an integer.