Density with flat Fourier transform around the origin

Question

I am looking for a density function f on $\mathbb{R}^2$ (nonnegative function that integrates to 1) with compact support which Fourier transform is constant (equal to 1) on a neighbourhood of the origin: the requirements are

• $f\geq 0$
• $\int_{\mathbb R^2} f=1$
• $f(x)=0$ for $\|x\|\geq R$ for some $R>0$
• $\hat f(x)=1$ for $x\in B(0,\epsilon)$ for some $\epsilon>0$

Equivalently I was looking for $g$ square integrable such that $g\star g=1$ around the origin, but not more luck on this one...

If you have an example on $\mathbb R^d$ I am also interested.

Edit: $f$ can be a distribution, but not the Dirac mass in 0 :)

Answer 1

It doesn't exists. The Fourier transform of any distribution with compact support is analytic, but analytic functions cannot be constant on a set with an acummulation point unless they are constant, but in that case $f$ would be a dirac mass, which you don't allow.