Let $f$ be a compact supported smooth function in $\mathbb{R}^n$, with compact support in $K$. We can define its Fourier transform as $$\widehat{f}(\xi)=\int_{\mathbb{R}^n}f(x)e^{-ix\cdot\xi}~dx$$ and its inverse Fourier transform as
$$f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^n}\widehat{f}(\xi)e^{ix\cdot\xi}~d\xi.$$
If we were to write $\mathbb{R}^n$ as two disjoint sets $\Omega_1$ and $\Omega_2$, then the function $f$ may be written as a sum
$f=f_1+f_2$, where
$$f_1(x)=\frac{1}{(2\pi)^d}\int_{\Omega_1}\widehat{f}(\xi)e^{ix\cdot\xi}~d\xi,~\&$$
$$f_2(x)=\frac{1}{(2\pi)^d}\int_{\Omega_2}\widehat{f}(\xi)e^{ix\cdot\xi}~d\xi.$$
What, if anything, can be said about the supports of $f_1$ and $f_2$? Would they be contained in the support of $f$? What if $f$ is positive? Would $f_1$ and $f_2$ be positive too?