ABSTRACT
The Fourier transform is an integral transform. Except for a very few functions whose Fourier transforms can be evaluated analytically, we can only find the Fourier transform F(s) of f(x) by performing a numerical integration [i.e., a Fast Fourier transform (FFT)]. Because FFT’s work best with evenly sampled data, let us assume that we have measured some f(x) between x=0 and x =xmax. Suppose we take N points xn, n=0,…, N-1, and sample f(x) at intervals dx=xmax/N. How large should N
be? How many samples do we need of f(x)? We can also switch the question around: how large should dx be to ensure that we retain in F(s) all of the information that was in f(x)? If we use an interval dx that is too large (N too small), we will distort F (s) . I shall show what the value of dx should be by comparing the Fourier transform to the Fourier series representation of f(x) .
