Functions with Fourier support in $[0,\infty)$.

Question

The Paley-Wiener space is given by class of functions $f \in L^2(\mathbb R)$ with the property that $$ \hat f(\xi) = 0, \quad |\xi| > \tfrac{1}{2}, \quad \hat f(\xi) = \int_{\mathbb R} f(x)e^{-2\pi i x \xi} dx $$ Hence, this class consists of all functions with compact support in $[-0.5,0.5]$ in Fourier space. Now let $H$ be the class of functions which have Fourier support in a half-space, say $[0,\infty)$. Does this class have a special name or is it studied somewhere? I couldn't find much on google.