Given a set of complex numbers $\mathcal A$, is there a convenient solution for the Fourier transform of its indicator function $\chi_{\mathcal A}(z)$? More specifically, if $\mathcal A$ is a set of natural numbers and $A(x)$ is the number of elements of $\mathcal A$ less than or equal to $x$, can we express $\mathcal F[\chi_{\mathcal A}](\xi)$ in terms of $A(x)$ or some other metric of $\mathcal A$?
I'm a complete beginner with respect to Fourier series, so I apologize if this is something I should be able to solve by myself.
(Note: when tagging, I noticed that the characteristic-functions tag description mentions that random variables are the Fourier transform of characteristic functions, but I couldn't find anything about that online. Feeling very stupid. Explanation?)
Best Answer
The basic case is that the Fourier transform of the indicator function of an interval centered at zero is a sinc function. So by this calculation and linearity, your question essentially reduces to determining how the Fourier transform is affected by translation. It is easy to calculate this by a change of variable: you find that $\widehat{f(x-x_0)}(\xi) = e^{-i x_0 \xi} \hat{f}(\xi)$. (As usual, everything is slightly different if you use a different normalization.) So the Fourier transform of a sum of indicator functions of intervals is a sum of sincs, each multiplied by $e^{-i x_k \xi}$ where $x_k$ is the center of the $k$th interval.
On characteristic functions of random variables: the characteristic function of a random variable is the Fourier transform of its density (in the sense of distribution theory, if the variable isn't continuous).