How to re-scale and do correctly the discrete Fourier transform.

I'm stuck with the re-scaling and the proper choice of parameters in doing the discrete Fourier transform. I explain: Suppose you want to calculate the Fourier transform

$$F(p) = \frac{1}{2\pi}\int dx e^{-ipx} f(x)$$

numerically by using Fast Fourier Transform (FFT). My function either in $$x$$ or $$p$$ is well localized, meaning that $$f(x)$$ is localized in a certain region on $$(x,p)$$ (for instance it is a gaussian). This would allow me to evauate the integral from an $$x_{min}$$ to an $$x_{max}$$ numerically. By discretizing the integral I get:

$$F(p_k)\sim \frac{1}{2\pi}\sum_n e^{-ip_k x_n} f(x_n)\Delta x,$$

where $$x_{n} = \Delta x*n + x_{min}$$. Thus

$$F(p_k)\sim \frac{1}{2\pi}e^{-ip_kx_{min}}\sum_ne^{-ip_k\Delta x n}f(x_n)\Delta x.$$

At this point, I want to calculate the FFT, and I make the identification $$p_k*\Delta x*n = 2\pi kn/N$$ ($$N$$ is just the total number of terms evaluated in the sum).

Questions:

1. How do one chooses $$\Delta x$$?

2. The FFT will give me the function at the values of $$p_k=2\pi k/\Delta xN$$, for $$k\in[0,1,\dots,N-1]$$. What is the proper re-scaling such that $$\lbrace p_k\rbrace$$ ranges in the same interval as $$x_n$$ (i.e. that $$p_{min} = x_{min}$$ and $$p_{max} = x_{max}$$)?

• The interval where you are approximating $F$ quite well can't be the same as the one where you discretized $f$. It seems you have almost all the clues. Note $|e^{-i (p+b) x}-e^{-ipx} | \le 2 |bx|$ so you should start from the functions $h(a)= \int_{|x| > a} (1+|x|)|f(x)|dx$ and $\sup_x |f'(x)|$ to find the interval and sample rate needed to approximate $F$ on some interval up to precision $\epsilon$. – reuns Feb 9 at 19:11