Can you construct a smooth indicator function $f$ on a set such that $\|\hat{f}\|_{L^1}\leq C$ for some given $C > 0$?

Question

Suppose our Fourier transform is given by $\mathcal{F}(f)(\xi) := \int_{\mathbb{R}^n}e^{-i\theta\left<x,\xi\right>}f(x)dx$ for some positive $\theta > 0$. I was wondering how/if you can construct a smooth indicator function on a set such that its Fourier transform's $L^1$ norm is at most, or equal, to some given constant? I am quite aware of the well-known smooth bump function, and the process of mollifying an indicator function of a set to get a smooth indicator function equal to one on the set and vanishing outside a compact set. But I don't really know how to confirm/deny that if $U\subset \mathbb{R}^n$ is any set, and $C > 0$ is given, whether we can construct a smooth indicator function $f$ such that $\|\hat{f}\|_{L^1(\mathbb{R}^n)}\leq C$; better yet if $\|\hat{f}\|_{L^1(\mathbb{R}^n)} = C$. All hints/pointers/solution attempts are welcome!