Fourier integral/ Fourier transformation of an oscillatory function with FFT

Question

f(x) = cos(x^2) and g(k) = pi^0.5 cos((pi*k)^2 - pi/4) are a Fourier pair.

I want to reproduce g(k) by Fourier integrating f(x) using FFT, i.e.

approximating Integrate[ f(x) * exp(2 pi * ikx), {x, -inf, inf} ]

with Sum[ fn * exp(2 pi * ik x_n), {n, 0, N-1} ] * Delta_x

However the result agrees with g(k) only on very small k ranges if it agrees at all (the same code works well for smooth Fourier pairs e.g. the Gaussian functions). I guess the problem is choosing appropriate values for N and Delta_x. Are there any established rules for how to choose them? Where can I find related topics in literature (I've read Numerical Recipe section 13.9 but it does not seem to solve my problem)?

Answer 1

An FFT inherently windows f(x) with a rectangular window of length N. This will produce different g(k) results for any k that is not exactly periodic within that window. You can try picking Delta_X and N so that their result is an exact integer multiple of your desired k.