Show that we can find a function $f \in L^1(\mathbb R^n)$ s.t. the Fourier transform of $f$, $\hat{f}=1$ in some neighborhood of $0.$

Question

How can we find a function $f \in L^1(\mathbb R^n)$ s.t. the Fourier transform of $f$, $\hat{f}=1$ in some neighborhood of $0$?

Answer 1

1. Using bump functions, you can construct a compactly supported, smooth function that equals $1$ in some neighborhood of $0$.
2. The Fourier transform is a bijection on the space of radily decreasing functions.