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It is well-known that on $\mathbb{R}$, the delta function $\delta(x)$ has the Fourier transform representation \begin{equation} \delta(x)=\int_\mathbb{R} e^{i2\pi kx}dk \end{equation}

which holds in the sense of distributions.

Let us think of a finite lattice $\{ -\frac{1}{2}, \cdots, 0 , \cdots, \frac{1}{2} \}$ with the spacing $\frac{1}{N}$ for some large $N \in \mathbb{N}$, so that the lattice has $2N+1$ elements.

Then, what would be discrete Fourier representation of the Kronecker delta function $\delta_{0, x}$? I guess it would be a discrete sum with a factor $\frac{1}{N}$ multiplied, but cannot figure out an exact form.

Also, in what sense does this discrete Fourier transform hold?

Could anyone clarify in the case of discreteness?

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For any $n=0,\dots,N-1$

$$\frac 1 N \sum_{k=0}^{N-1}e^{2i\pi \frac{kn}N} = \delta_{0,n}\tag{1}$$ This is the discrete equivalent to the Poisson summation formula or the formula you wrote. So the Kronecker delta is discrete equivalent of the Dirac delta and $(1)$ is saying that the Discrete Time Fourier Transform of the constant function is that delta (of course, you may define the DTFT with a different normalization factor).

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