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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 $p_{max} = x_{max}$)?

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  • $\begingroup$ 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 $\epsilon$. $\endgroup$ – reuns Feb 9 at 19:11

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