Let $x,y,z,...$ be $N$ points and considerer the Kronecker delta associated to the sequence, i.e. $\delta_{x,y} = 1 $ if and only if $x=y$.
My definition of Discrete Fourier Transform is: the "discrete Fourier transform" transforms a sequenceof $N$ complex numbers $x_0, x_1,..., x_{N-1}$ into another sequence of complex numbers, $X_0, X_1, \ldots, X_{N-1}$ which is defined by $$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-\frac {i 2\pi}{N}kn}$$
So, $\delta_{x,y}$ takes the place of $x_n$ in the last formula, and the sum goes from $0$ to $N^2$. Now, because the definition of Kronecker delta, I found that for the Fourier coefficient holds the following
$$X_k=\sum_{n=0}^{N-1}e^{-i \frac{2\pi}{N^2}kn}$$.
in view of the fact that in general one has $N^2$ points and that $\delta_{x,x}=1$.
The problem is: what I expected is that the Fourier coefficient $X_k$ was equal to $1$ for every $k$.
What is wrong?