Inverse Fourier transform of characteristic function

Question

Let $Q$ be a measurable set in $\mathbb{R}^2$ Let \begin{equation} 1_Q(\textbf{x}) = \left\{ \begin{array}{ll} 1 & \mbox{if $\textbf{x} \in Q$},\\ 0 & \mbox{otherwise},\end{array} \right. \end{equation} I am looking for an expression for the inverse Fourier transform \begin{equation} \mathcal{F}^{-1}1_Q(\textbf{u})=h(u_1,u_2). \end{equation} Is it possible to write it in terms of a 2D $\mbox{sinc}$ function?

Answer 1

Answering the follow-up question in comments: Yes, the Fourier transform on 2D has certain scaling properties, explained here. The difference is that the geometry in dimensions above $1$ is much richer. In one dimension, scaling and translation suffice to transform any connected set into a standard interval such as $[-1,1]$. In $\mathbb R^2$ we have infinitely many kinds of shapes, which can not be catalogized even up to scaling/translation. For example:

• Fourier transform of a rectangle is the product of two $\mathrm{sinc} $ functions.
• Fourier transform of a disk is a Bessel function, an entirely different animal.