Frames of multi-windowed exponentials on subsets of ${\mathbb{R}}^d$

Given discrete subsets ${\mathrm{\Lambda }}_j\subset {\mathbb{R}}^d$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}{g_j(x)e^{2\pi i <\lambda,x>}: \lambda\in\Lambda_j}$on$L^2(\Omega)$. We show that a necessary and sufficient condition for the windows $g_j$ to form a frame of windowed exponentials for $L^2(\mathrm{\Omega })$ with some ${\mathrm{\Lambda }}_j$ is that $m\leq \max_{j\in J}|g_j|\leq M$almosteverywhereon$\Omega$forsomesubset$J$of${1,…, q}$\mathrm{.}If$\Omega$ is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of $\mathrm{\Omega }$ is infinite. If $\mathrm{\Omega }$ is unbounded but of finite measure, we give a sufficient condition for the existence of Fourier frames on $L^2(\mathrm{\Omega })$. At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.