For the discrete Fourier transform, it is defined by
$$f(k)=\sum_{s_i}\exp(-iks)\phi(s).~~~~~~~~~~~~~~~~~(ds-1)$$ here $s_i=-(N-1)/2,-(N-2)/2,.....(N-1)/2$.
For convenience, we also add the continuous Fourier transform
$$f(k)=\int_{-\infty}^{\infty}\exp(-iks)\phi(s)ds~~~~~~~~~~~~~~~~~(cn-1)$$
It can be seen that for the discrete case, the integral of the right-hand side of Eq.~(cn-1) is chosen at some special point, i.e., $s_i=-(N-1)/2,-(N-2)/2,.....(N-1)/2$. If you take $N\to \infty$, the above two formulas should consistent with each other.
While for the inverse discrete Fourier transform, it reads
$$\phi(s)=\frac{1}{N}\sum_{k_i}\exp(iks)f(k),~~~~~~~~~~~~~~~~~(ds-2)$$ where $k_i=-\frac{2\pi}{N}\frac{N-1}{2},......\frac{2\pi}{N}\frac{N-1}{2}$. Let $N\to \infty$, Eq.~(ds-2) can be shown by $$\phi(s)=?\frac{1}{N}\int_{-\pi}^{\pi}\exp(iks)f(k)dk~~~~~~~~~~~~~~~~~(ds-3)$$ I believe Eq.~(ds-3) is wrong if you let $N\to \infty$.
I see that in some books, they use
$$\phi(s)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\exp(iks)f(k)dk~~~~~~~~~~~~~~~~~(ds-4)$$
Question 1): how to understand the equation (ds-3) and (ds-4). Why $N$ should be replaced by $2\pi$?
Question 2): If N is finite, we can use Eq.~(ds-1) for the discrete Fourier transform, and the inverse is given by (ds-2). While, for the discrete case, if $N\to \infty$, how can I get the inverse discrete Fourier transform? Can we use Eq.~(ds-2)? But If $N\to \infty$, it seems that Eq.~(ds-2) is not correct.
Question 3) If $N\to \infty$, can we use the continuous Fourier transform, i.e., $$\phi(s)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(iks)f(k)dk~~~~~~~~(cn-2)$$ to estimate Eq.~($ds-2$)? It seems that Eq.($ds-4$) is different from Eq.~(cn-2). Is there some relation between Eq.~(ds-4) and Eq.(cn-2).
Question 4, whih formula is the discrete inverse Fourier tansfom in the limit of $N\to \infty$?
Any suggestions or related URL or books are welcome! Thanks!