This may be an elementary question, but I don't know much about the discrete Fourier transforms (DFT).
Suppose I have a sequence $\{x_n\}_{n=0}^{N-1}$ of $n$ real numbers such that $x_0\geq |x_n|$ for all $n$. I take the DFT of the sequence as follows: $X_k=\sum_{n=0}^{N-1}x_n e^{-i2\pi\frac{k}{N}n}$.
I am wondering if the frequency-domain representation of this sequence has any special properties besides the symmetry associated with real-valued input. In particular, I am wondering whether anything useful can be said about the real parts of $\{X_k\}$ (s.t. whether they are positive).