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!