What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,

What properties are common to Fourier transforms of all characteristic functions?

Here are a few trivial properties; what other properties are known?

• $\widehat{1_A}$ is a bounded continuous function converging to zero.
• $\widehat{1_A}$ is in $L^2$.

What interesting functional analytic properties does the set $\{\widehat{1_A}: A \in \mathbb{R}^n\}$ of Fourier transforms of characteristic functions have?

At least it is closed in $L^2$ (just apply Plancherel's formula and use the fact that an $L^2$-limit of characteristic functions is a characteristic function). Is it closed in other norms? Is it dense in some interesting spaces (if we are allowed to multiply the functions by a constants)? For which $p$ is the Fourier transform a bounded operator from our set to $L^p$?

How does the regularity (in a vague sense) of $A$ affect on the decay of $\widehat{1_A}$? Are these Fourier transforms always entire?

Here are a few easy remarks:

• If $A$ is a finite union of intervals, then $\hat{1}_A(\xi)$ is a trigonomteric polynomial divided by $\xi$, so it is in every $L^p,p>1$ but not absolutely integrable.
• If $A$ is bounded, the Fourier transform is an entire function in $\mathbb{C}^n$.