1) It is known that no function vanishing outside an interval $-a < x < a $ has a Fourier transform with compact support. $(x,a \in \mathbb{R})$ However, are there Fourier transform pairs which both vanish inside the interval $-a < x < a$, and are Fourier transform eigenfunctions with this property?
In other words, for all $f(x)$: $$ \int_{-\infty}^{\infty} f(x) \Pi (\frac{x}{a})e^{-i2 \pi xy} \, dx \neq \lambda f(x) \Pi (\frac{x}{a})$$ However, are there functions, $g(x)$, solving this?: $$ \int_{-\infty}^{\infty} g(x) (1-\Pi (\frac{x}{a}))e^{-i2 \pi xy} \, dx = \lambda g(x) (1-\Pi (\frac{x}{a}))$$
$N.B.$ - $\Pi (\frac{x}{a})$ is the $rect(x)$ function scaled to a width of $a$.
2) Are there keywords or names for this topic, that I might do more research? $i.e.$ Is there a specific name for this domain "outside an interval?"